Uni e sidade de San iago de Compos ela
Depa amen o de Es a ´ıs ica e In es igaci´on Ope a i a
Tes s in Nonpa ame ic Reg ession
Based on he E o Dis ibu ion
Juan Ca los Pa do Fe n´andez
Tes s in Nonpa ame ic Reg ession
Based on he E o Dis ibu ion
Juan Ca los Pa do Fe n´andez
Realizado o ac o p´ublico de de ensa e man emen o des a ese dou o al o d´ıa 1 de
decemb o de 2005 na Facul ade de Ma em´a icas da Uni e sidade de San iago de
Compos ela, pe an e o ibunal o mado po
P esiden e D . D. Rica do Cao Abad
Vogais D . D. No¨el Ve a e beke
D . D. F ´ed´e ic Fe a y
D . D. Jacobo de U˜na ´
Al a ez
Sec e a io D . D. C´esa And ´es S´anchez Selle o,
sendo os di ec o es D . D. Wenceslao Gonz´alez Man eiga e D a. Dna. Ing id
Van Keilegom, ob i o a m´axima cuali icaci´on de sob esalien e cum laude.

Ded´ıcolles es e aballo
aos meus pais e i maus
Ag adecemen os
Que o comeza es a p´axina exp esando o meu ag adecemen o aos p o eso es
Wenceslao Gonz´alez Man eiga e Ing id Van Keilegom polo labo de di ecci´on des e
aballo. I am e y g a e ul o Ing id o in i ing me o isi he Ins i u de
S a is ique a Lou ain-la-Neu e and accep ing me as a doc o a e s uden . I ha e
lea ned a lo om bo h o hem. Moi as g azas, danke je.
Gus a ´ıame am´en da lles as g azas aos compa˜nei os do Depa amen o de Es-
a ´ıs ica e In es igaci´on Ope a i a da Uni e sidade de Vigo que me apoia on ao
longo dos ´ul imos anos. Desexo menciona exp esamen e a Jacobo e a Gus a o
polo in e ese mos ado ca a ao colec i o dos p o eso es isi an es. Con moi os
deles ´uneme, ademais dunha elaci´on p o esional, unha g ande amizade: Albe o
e Ja ie ( an as ho as compa idas ´o a e den o do despacho), Ma ´ıa, Le icia,
Amalia, Sil ia. Ag ad´ezolles a odos os bos momen os.
Une g ande pa ie de ce e h`ese a ´e ´e ´ealis´ee pendan plusieu s s ages de
eche che dans l’Ins i u de S a is ique de l’Uni e si ´e ca holique de Lou ain (Bel-
gique). Je oud ais eme cie ou le pe sonnel de l’Ins i u (mes coll`egues jeunes
che cheu s, p o esseu s e pe sonnel adminis a i ) pou le o midable accueil. Ils
on ai que le emps que j’ai pass´e `a Lou ain-la-Neu e ai ´e ´e `es ag ´eable. Un
g and me ci `a Bianca, Anoua , Agus ´ın e Oana pou leu ami i´e e pou m’a oi
aid´e plusiue s ois a ec mes p obl`emes de logemen .
Finalmen e, ago cons a que es e aballo oi inanciado polo Minis e io de
Ciencia y Tecnolog´ıa (p oxec o BFM2002-03213, que incl´ue ondos FEDER), pola
Sec e a ´ıa Xe al de In es igaci´on da Xun a de Galicia a a ´es da con oca o ia de
axudas pa a es ad´ıas en cen os de in es igaci´on e polo Vice ei o ado de In es i-
gaci´on da Uni e sidade de Vigo.
To ´es, xullo de 2005
Juan Ca los Pa do Fe n´andez
In oduc ion 5
unde he null hypo hesis. These es ima ed esiduals mus be conside ed as cen-
so ed da a because hey depend on he censo ed obse a ions Zij. The compa i-
son o he dis ibu ions o he esiduals is now ca ied ou wi h he Kaplan-Meie
es ima o (Kaplan and Meie , 1958), which is he equi alen o he empi ical dis-
ibu ion unc ion in he censo ed model. Theo e ical esul s and simula ions a e
also included.
The p oblem o es ing o he equali y o nonpa ame ic eg ession cu es has
been ea ed in he li e a u e du ing he las i een yea s, mainly in mo e es ic-
i e si ua ions. Some au ho s, such as Young and Bowman (1995) and De e and
Neumeye (2001), named his p oblem ‘Nonpa ame ic Analysis o Co a iance’.
The i s e e ences abou compa ison o eg ession cu es assume some es ic-
ions on he model. Fixed design and homoscedas ic e o s a e common assump-
ions in se e al e e ences. H¨a dle and Ma on (1990), Hall and Ha (1990), King,
Ha and Weh ly (1991), and Delgado (1993) assume equal design poin s, equal
sample sizes and homoscedas ic e o s. H¨a dle and Ma on (1990) in oduce a
semipa ame ic me hod o es he hypo hesis o equali y o wo eg ession cu es
by pa ame e izing he di e ence be ween hem. Hall and Ha (1990) wo k wi h
boo s ap me hods. Delgado (1993) p esen s an app oach based on he empi ical
p ocess o he di e ence o he esponses, which does no equi e smoo hing a all.
Young and Bowman (1995) use he classical idea o analysis o co a iance in o de
o de elop a me hod o compa e mo e han wo cu es, assuming homoscedas ic
and no mal dis ibu ed e o s. The same au ho s in Bowman and Young (1996)
p esen a g aphical me hod o check he equali y o eg ession cu es. Kulaseke a
(1995) and Kulaseke a and Wang (1997) p opose a es applicable in he case o
unequal design poin s, bu i s ill equi es homoscedas ic e o s and he impac o
he choice o he smoo hing pa ame e is impo an and canno be a oided in an
easy way.
Munk and De e (1998) conside he p oblem o es ing o he equali y o
wo cu es wi h he e oscedas ic e o s and ixed design. The es ing p ocedu e is
based on he es ima ion o he L2-dis ance o he di e ence be ween he wo cu es.
Scheike (2000) es ablishes a es by compa ing wo cumula i e eg ession unc ions

6 Chap e 1
and ex ends he ideas o Delgado (1993) o di e en and andom designed co a i-
a es. De e and Neumeye (2001) desc ibe se e al me hods based on smoo hing
echniques o compa e mo e han wo cu es in a nonpa ame ic amewo k wi h
ixed design. Kulaseke a and Wang (2001) apply he ideas o he likelihood a io
es s o compa e wo eg ession cu es. Finally, Neumeye and De e (2003) use
an empi ical p ocess app oach in a comple e nonpa ame ic, he e oscedas ic and
andom designed se up o compa e wo eg ession cu es. Thei es enjoys he ca-
pabili y o de ec ing al e na i es con e ging o he null hypo hesis a a a e n−1/2,
which is o in e es .
A di e en app oach consis s o es ing he equali y o wo eg ession cu es
agains mo e speci ied al e na i es. The con ibu ions o Hall, Hube and Speck-
man (1997), Koul and Schick (1997, 2003), Neumeye and De e (2005) and Speck-
man, Chin, Hewe and Be elson (2003) deal wi h one-sided al e na i es, which is
an impo an pa icula si ua ion. In his case he null hypo hesis is he equali y
o he cu es and he al e na i e says ha one cu e is s ic ly bigge han he
o he one. Fe ei a and S u e (2004) also wo k wi h he ideas o Delgado (1993) in
a ime se ies con ex , and Vila -Fe n´andez and Gonz´alez-Man eiga (2004) conside
dependen e o s.
As can easily be seen, mos o hese e e ences a e de o ed o es ing o he
equali y o only wo eg ession cu es. Thei ex ensions o mo e han wo cu es
a e no s aigh o wa d. In p ac ical si ua ions he p oblem o es ing o he
equali y o mo e han wo eg ession cu es can a ise e y easily. I he compa ison
is pe o med pai wise, a co ec ion in he le el o he es s mus be done, and
consequen ly he powe can be a ec ed. This mo i a es he implemen a ion o
gene al p ocedu es o mo e han wo cu es, as has been done in his hesis.
To he bes o ou knowledge, he p oblem o es ing o he equali y o nonpa a-
me ic eg ession cu es wi h censo ed da a has no been ea ed in he li e a u e.
In Chap e 2 we also p opose a me hod o es o he equali y o he dis i-
bu ions o he e o s in k eg ession models. Conside he same s a is ical se up
such as in he compa ison o eg ession cu es p oblem. We es o he hypo hesis
H0ε:Fε1=Fε2=. . . =Fεk
In oduc ion 7
e sus he gene al al e na i e o any di e ence be ween hese dis ibu ions. The
equali y o he dis ibu ions o he e o s is a ela i ely common assump ion in some
e e ences abou compa ison o eg ession cu es, al hough i is no necessa y o
ou me hod.
Pa ame ic eg ession models a e appealing in many si ua ions. They desc ibe
he ela ionship be ween he esponse a iable and he co a ia e in a simple way
and usually allow o in e p e abili y o he pa ame e s ( o ins ance in linea
eg ession). Ne e heless, i he pa ame ic model ails hen he conclusions will
be e oneous. This mo i a es he de elopmen o speci ic goodness-o - i es s o
check he alidi y o pa ame ic models.
Gi en model (1.1) and a pa ame ic amily o eg ession unc ions
M={mθ;θ∈Θ},
wi h Θ ⊂Rp, we a e in e es ed in es ing o he null hypo hesis
H0:m∈ M
e sus he gene al al e na i e
H1:m /∈ M.
In Chap e 4 we s udy a goodness-o - i es o pa ame ic models when he
esponse a iable is subjec o igh censo ing. The goodness-o - i p ocedu e con-
sis s o compa ing wo es ima o s o he dis ibu ion o he e o s. Le (Xi, Zi,∆i),
i= 1, . . . , n, be an i.i.d. sample o (X, Z, ∆). Assume ha ˆ
θis an es ima o o
he pa ame e unde he null hypo hesis and ˆmis a nonpa ame ic es ima o o
he eg ession unc ion. We compa e he dis ibu ion o he esiduals es ima ed in
a pa ame ic way
Zi−mˆ
θ(Xi)
ˆσj(Xi)
wi h he dis ibu ion os he esiduals es ima ed in a comple ely nonpa ame ic way
Zi−ˆm(Xi)
ˆσ(Xi).
8 Chap e 1
These esiduals a e conside ed as censo ed da a and we compa e he Kaplan-Meie
es ima o s o hei dis ibu ions ia Kolmogo o -Smi no and C am´e - on Mises
ype s a is ics. Asymp o ic esul s will be s a ed. The c i ical alues o he es s
a e ob ained by a boo s ap p ocedu e, which is s udied by means o simula ions.
Se e al goodness-o - i es s o comple e da a ha e been p oposed in he li e a-
u e du ing he las wo decades. De e and Munk (1998), De e (1999) and De e,
Munk and Wagne (2000) es ima e he minimum L2-dis ance be ween mand he
pa ame ic amily o o he ela ed quan i ies. H¨a dle and Mammen (1993) con-
side he L2-dis ance o he di e ence be ween he pa ame ic and nonpa ame ic
es ima o s o he eg ession unc ion. S u e (1997) and S u e, Gonz´alez-Man eiga
and P esedo-Quidimil (1998) p oposed a di e en app oach based on he ma ked
empi ical p ocess o he in eg a ed eg ession unc ion and i s boo s ap app oxi-
ma ion. Ramil-No o and Gonz´alez-Man eiga (2000) s udied me hods o i poly-
nomial models. Mo e ecen ly, De e and He zle (2004) s udied es s based on
p ocesses indexed by he bandwid hs.
Fo censo ed da a S u e, Gonz´alez-Man eiga and S´anchez-Selle o (2000) s u-
died a es based on he ma ked empi ical p ocess o he in eg a ed eg ession
unc ion. They assume independence be ween he esponse Yand he censo ing
a iable Cand ocus on he condi ional mean.
The no a ion is sel -con ained in each chap e .
1.2 Nonpa ame ic es ima ion in eg ession
In his sec ion we b ie ly desc ibe some es ima ion echniques in nonpa ame ic
eg ession. Le (X, Y ) e i y he eg ession model
Y=m(X) + σ(X)ε,
whe e m(x) = E(Y|X=x) is he condi ional mean, σ2(x) = V a (Y|X=x)
is he condi ional a iance and he e o εis independen o X. Le (Xi, Yi), i=
1, . . . , n, be nindependen obse a ions o (X, Y ). The objec i e o nonpa ame ic
In oduc ion 9
eg ession is o gi e an es ima e o he eg ession cu e mbased on he sample
wi hou assuming any pa ame ic model.
Smoo hing echniques es ima e he alue o he eg ession cu e a a ce ain
poin xby a e aging locally he alues o he esponse co esponding o he alues
o he co a ia e which a e close o x. A smoo h es ima o o m(x) is
ˆm(x) =
n
X
i=1
Wi(x, h)Yi,(1.2)
whe e Wi(x) is a sequence o weigh s depending on xand a smoo hing pa ame e ,
h. The smoo hing pa ame e plays a undamen al ole in any smoo hing p oce-
du e because i con ols he impo ance o a pa icula esponse acco ding o he
p oximi y o xo he co esponding co a ia e.
The choice o he weigh s sequence leads o di e en smoo hing me hods. In
his piece o esea ch we will wo k wi h ke nel es ima o s. Pa icula ly, Nada aya-
Wa son weigh s, in oduced simul aneously by Nada aya (1964) and Wa son (1964),
will be used o es ima e he eg ession unc ion m
Wi(x, h) = K((x−Xi)/h)
Pn
i0=1 K((x−Xi0)/h),(1.3)
whe e h > 0 is he smoo hing pa ame e o bandwid h and Kis a ke nel unc ion
(no mally, a symme ic densi y). The Nada aya-Wa son es ima o o he a iance
unc ion is
ˆσ2(x) =
n
X
i=1
Wi(x, h)Y2
i−ˆm2(x).
As men ioned abo e, he bandwid h hplays a undamen al ole in ke nel es-
ima ion. I his oo la ge he es ima ion ˆm(x) is biased and he esul ing cu e
o e smoo hed. I his oo small, hen he esul ing es ima o has la ge a iance
and i is unde smoo hed. The e a e se e al mechanisms o choose he bandwid h
pa ame e in an op imal way: c oss- alida ion, plug-in me hods, boo s ap ech-
niques. On he o he hand, he choice o he ke nel unc ion Kdoes no ep esen
a big impac on he es ima ion. Fo u he discussion abou he choice o hand
Kand o he opics in ke nel es ima ion see e.g. H¨a dle (1990), Wand and Jones
(1994) o H¨a dle, M¨ulle , Spe lich and We wa z (2004).
10 Chap e 1
The local polynomial me hods, s udied in Fan and Gijbels (1996), cons i u e
an ex ension o he Nada aya-Wa son es ima o . Al e na i e choices o he weigh s
sequence in (1.2) include Gasse and M¨ulle weigh s and k-nea es neighbo weigh s
(see H¨a dle, 1990). O he smoo hing echniques a e spline me hods and o hogonal
se ies es ima o s (see Ha , 1997).
Ak i as and Van Keilegom (2001) s udied he p ope ies o he ollowing non-
pa ame ic es ima o o he dis ibu ion o he e o s
ˆ
Fε(y) = 1
n
n
X
i=1
IµYi−ˆm(Xi)
ˆσ(Xi)≤y¶,(1.4)
which is he empi ical dis ibu ion o he es ima ed esiduals, whe e ˆmand ˆσa e
he Nada aya-Wa son es ima o s o mand σ.
Assume now ha he esponse a iable Yis subjec o igh censo ing. This
means ha he e exis s a andom a iable C, independen o Ygi en X, such ha
ins ead o Ywe obse e he pai (Z, ∆), whe e Z= min{Y, C}and ∆ = I(Y≤C).
We wo k wi h he ollowing de ini ions o he eg ession and a iance unc ions
m(x) = Z1
0
F−1(s|x)J(s)ds, (1.5)
σ2(x) = Z1
0
F−1(s|x)2J(s)ds −m2(x),(1.6)
whe e F−1(s|x) = in { ;F( |x)≥s}is he condi ional quan ile unc ion o Ygi en
X=xand Jis a gi en sco e unc ion sa is ying R1
0J(s)ds = 1.
In o de o es ima e he eg ession and he a iance unc ions nonpa ame-
ically, we need a nonpa ame ic es ima o o he condi ional dis ibu ion o Y
gi en X. Le (Xi, Zi,∆i), i= 1, . . . , n, be a sample o independen obse a ions
o (X, Z, ∆). The nonpa ame ic es ima o o he condi ional dis ibu ion wi h
censo ed da a was in oduced by Be an (1981)
ˆ
F(y|x) = 1 −Y
Zi≤y,∆i=1 µ1−Wi(x, h)
Pn
l=1 I(Zl≥Zi)Wl(x, h)¶,
whe e Wi(x, hn) a e he Nada aya-Wa son ype weigh s gi en in (1.3). This es i-
ma o was s udied, among o he s, by Dab owska (1989), Gonz´alez-Man eiga and

In oduc ion 11
Cada so-Su´a ez (1994) and Van Keilegom and Ve a e beke (1996, 1997). In ab-
sence o censo ing ˆ
F(y|x) educes o he es ima o o he condi ional dis ibu ion
p oposed by S one (1977). The igh ails o he Be an es ima o may be inconsis-
en o es ima e he condi ional dis ibu ion due o he censo ing s uc u e. This
mo i a es he ede ini ion o he eg ession and a iance unc ions o include he
sco e unc ion J.
F om equa ions (1.5) and (1.6), i is clea ha once we ha e an es ima o o he
condi ional dis ibu ion unc ion, we can immedia ely cons uc an es ima o o
he eg ession unc ion. Nonpa ame ic es ima o s o he eg ession and a iance
unc ions a e gi en by
ˆm(x) = Z1
0
ˆ
F−1(s|x)J(s)ds,
and
ˆσ2(x) = Z1
0
ˆ
F−1(s|x)2J(s)ds −ˆm2(x).
The esiduals o he eg ession model can now be es ima ed by
ˆ
Ei=Zi−ˆm(Xi)
ˆσ(Xi),
conside ed as censo ed quan i ies since hey a e measu ed wi h espec o he
obse able a iables Ziand no wi h espec o he ue (and no obse able)
esponses Yi. Van Keilegom and Ak i as (1999) p oposed es ima ing he dis ibu-
ion o he e o s by he Kaplan-Meie es ima o o he censo ed sample ( ˆ
Ei,∆i),
i= 1, . . . , n,
ˆ
Fε(y) = 1 −Y
ˆ
Ei≤y,∆i=1 Ã1−1
Pn
l=1 I(ˆ
El≥ˆ
Ei)!.
Ob iously, his es ima o o he dis ibu ion o he e o s educes o he one gi en
in (1.4) in absence o censo ing.
Chap e 2
Tes ing o he equali y o k
eg ession cu es
In his chap e we in oduce a p ocedu e o es he hypo hesis o equali y o
he k eg ession unc ions in a comple ely nonpa ame ic amewo k. The es
is based on he compa ison o wo es ima o s o he dis ibu ion o he e o s in
each popula ion. Kolmogo o -Smi no and C am´e - on Mises ype s a is ics a e
conside ed and hei asymp o ic dis ibu ions a e ob ained. The p oposed es s
can de ec local al e na i es con e ging o he null hypo hesis a he a e n−1/2.
We desc ibe a boo s ap p ocedu e in o de o app oxima e he c i ical alues,
and p esen he esul s o a simula ion s udy, in which he beha io o he es s o
small and mode a e sample sizes is s udied. Finally, we include an applica ion o
a eal da a se . As a by-p oduc , a me hod o es he equali y o he dis ibu ion
o he e o s in se e al eg ession models is ob ained and de eloped heo e ically
and om a p ac ical poin o iew.
This chap e is mainly based on Pa do-Fe n´andez, Van Keilegom and Gonz´alez-
Man eiga (2005a).
13
14 Chap e 2
2.1 In oduc ion. S a is ical model
The compa ison o wo o mo e g oups is an impo an p oblem in s a is ical
in e ence. This compa ison can be pe o med h ough he eg ession cu es in a
nonpa ame ic con ex . Le (Xj, Yj) be kindependen andom ec o s, and assume
ha hey sa is y he ollowing nonpa ame ic eg ession models, o j= 1, . . . , k,
Yj=mj(Xj) + σj(Xj)εj,(2.1)
whe e he e o a iable εj, wi h dis ibu ion Fεj, is independen o Xj,
mj(Xj) = E(Yj|Xj)
is he unknown eg ession unc ion and
σ2
j(Xj) = V a (Yj|Xj)
is he condi ional a iance unc ion. By cons uc ion E(εj) = 0 and V a (εj) = 1.
Suppose ha he co a ia es Xjha e common suppo RX. Le (Xij, Yij), i=
1, ..., nj, be an i.i.d. sample om he dis ibu ion o (Xj, Yj), o j= 1, . . . , k, and
deno e n=Pk
j=1 nj.
We a e in e es ed in es ing he null hypo hesis o equali y o he eg ession
unc ions
H0:m1=m2=··· =mk(2.2)
e sus he gene al al e na i e
Ha:mi6=mj o some i, j ∈ {1,2, . . . , k}.
The idea o he es ing p ocedu e p oposed in his hesis is o compa e, in each
popula ion, he empi ical dis ibu ion unc ions o he esiduals wi h he same
dis ibu ion unc ion es ima ed unde he null hypo hesis (2.2). Mo e p ecisely,
le us ix one popula ion, say j. Le
Yij −ˆmj(Xij)
ˆσj(Xij)
Tes ing o he equali y o k eg ession cu es 21
2.3.1 Asymp o ic esul s unde he null hypo hesis
In his sec ion we s a e he asymp o ic esul s ela ed o he es ing p ocedu e
unde he null hypo hesis. Theo em 2.2 gi es a ep esen a ion o he di e ence o
he wo es ima o s o he e o dis ibu ion in each popula ion ˆ
Fεj0(y)−ˆ
Fεj(y) as
a sum o i.i.d. andom a iables plus a negligible e m, which will be used in Theo-
em 2.3 o s a e he weak con e gence o he k-dimensional p ocess ˆ
W(y). Finally,
he asymp o ic dis ibu ions o he es s a is ics a e ob ained in Co olla y 2.1.
Theo em 2.2 Assume (A1)-(A4). Then, unde he null hypo hesis H0, o j=
1, . . . , k
ˆ
Fεj0(y)−ˆ
Fεj(y)
= εj(y)
k
X
l=1
pl(1
nl
nl
X
i=1
Yil −m(Xil)
σj(Xil)µ Xj(Xil)
mix(Xil)−I(l=j)
pj¶)+oP(n−1/2),
uni o mly in y.
Theo em 2.3 Assume (A1)-(A4). Then, unde he null hypo hesis H0, he k-
dimensional p ocess ˆ
W(y)=(ˆ
W1(y), . . . , ˆ
Wk(y)) con e ges weakly o W(y) =
( ε1(y)W1, . . . , εk(y)Wk) , whe e, W1, . . . , Wka e no mal andom a iables wi h
mean ze o and co a iance s uc u e
Co (Wj, Wj0) = p1/2
jp1/2
j0×
×
k
X
l=1
plE·σ2
l(Xl)
σj(Xl)σj0(Xl)µ Xj(Xl)
mix(Xl)−I(l=j)
pj¶µ Xj0(Xl)
mix(Xl)−I(l=j0)
pj0¶¸.
Co olla y 2.1 Assume (A1)-(A4). Then, unde he null hypo hesis H0,
T1
KS
d
→
k
X
j=1 |Wj|sup
y| εj(y)|,
T1
CM
d
→
k
X
j=1
W2
jZ 2
εj(y)dFεj(y),
T2
KS
d
→sup
y|W(y)|,
T2
CM
d
→ZW2(y)dFε(y),

22 Chap e 2
whe e W(y) = Pk
j=1 p1/2
j εj(y)Wjand Fε(y) = Pk
j=1 pjFεj(y).
2.3.2 Asymp o ic esul s unde local al e na i es
Conside now he limi ing beha io o he es s a is ics unde he ollowing local
al e na i es:
Hl.a. :mj=m0+n−1/2 j,
whe e he unc ions jsa is y
(A5) (i) jis wice con inuously di e en iable, o j= 1, . . . , k.
(ii) V a [ j(Xl)] <∞, o j= 1, . . . , k and l= 1, . . . , k.
In Theo em 2.4 we s a e he weak con e gence o ˆ
W(y) unde he al e na-
i e hypo hesis Hl.a. and in Co olla y 2.2 we gi e he co esponding asymp o ic
dis ibu ions o he es s a is ics.
Theo em 2.4 Assume (A1)-(A5). Then, unde he al e na i e hypo hesis Hl.a.,
he k-dimensional p ocess ˆ
W(y) = ( ˆ
W1(y), . . . , ˆ
Wk(y)) con e ges weakly o W(y)+
D(y), whe e W(y)is de ined in Theo em 2.3 and
D(y) = (p1/2
1 ε1(y)d1, . . . , p1/2
k εk(y)dk) ,
wi h
dj=E·R(Xj)− j(Xj)
σj(Xj)¸,
and R(u) = Pk
j=1 pj
Xj(u)
mix(u) j(u).
Co olla y 2.2 Assume (A1)-(A5). Then, unde he al e na i e hypo hesis Hl.a.,
T1
KS
d
→
k
X
j=1 |Wj+p1/2
jdj|sup
y| εj(y)|,
T1
CM
d
→
k
X
j=1
(Wj+p1/2
jdj)2Z 2
εj(y)dFεj(y),
T2
KS
d
→sup
y|W(y) + d(y)|,
T2
CM
d
→Z(W(y) + d(y))2dFε(y),
Tes ing o he equali y o k eg ession cu es 23
whe e d(y) = Pk
j=1 pj εj(y)dj, he andom a iables Wja e de ined in he s a e-
men o Theo em 2.3, and W(y)and Fε(y)a e de ined in he s a emen o Co ol-
la y 2.1.
We can analyze in de ail he e ec o he local al e na i es i we conside he
simple si ua ion o wo eg ession cu es whe e one o he cu es is ixed and he
o he one a ies wi h n. The null hypo hesis is H0:m1=m2and he al e na i e
Hl.a. :m2=m1+n−1/2 . In his si ua ion
d1=p2E· X2(X1)
mix(X1)
(X1)
σ1(X1)¸
and
d2=−p1E· X1(X2)
mix(X2)
(X2)
σ2(X2)¸,
and hese alues may be ze o in some cases. Ne e heless, he e a e impo an
si ua ions wi h consis ency agains al e na i es con e ging o he null hypo hesis
a a a e n−1/2, such as he one-sided al e na i es (when is a posi i e unc ion).
And, o cou se, he es ing p ocedu e is uni e sally consis en in he sense o
Theo em 2.1.
2.4 Boo s ap app oxima ion
To apply his es ing p ocedu e in p ac ice he asymp o ic dis ibu ion o he
es s a is ics can be used so as o ob ain he c i ical alues o he es . These
asymp o ic dis ibu ions, gi en in Co olla y 2.1, can be es ima ed by plugging
in es ima o s o pj, m, σj, Fεj, εj, Xjand mix. Al e na i ely, one can use a
boo s ap p ocedu e in o de o app oxima e he dis ibu ions o he es s a is ics
unde he null hypo hesis. We will now conside his second op ion in de ail.
Fi s , o j= 1, . . . , k and i= 1, ..., nj, es ima e he esiduals in a nonpa ame-
ic way, using each sample sepa a ely, ha is
Yij −ˆmj(Xij)
ˆσj(Xij).
24 Chap e 2
These esiduals a e hen s anda dized in o de o ha e mean ze o and a iance
one. Le ˜
Fεjbe he empi ical dis ibu ion o hese s anda dized esiduals ob ained
om each sample.
We p opose a smoo h boo s ap o he esiduals. No e ha he asymp o ic
ep esen a ion gi en in Theo em 2.2 in ol es he densi y o he e o s εj. This
sugges s ha a smoo hed e sion o he boo s ap o he esiduals mus be used.
In he boo s ap o he esiduals he samples a e d awn om he empi ical dis ib-
u ion, while in he smoo h boo s ap he esamples a e d awn om an es ima ion
o he co esponding densi y. See F eedman (1981) o he boo s ap o he esi-
duals and, e.g., Da ison and Hinkley (1997) o Sil e man and Young (1987) o
he smoo hing in he boo s ap.
The boo s ap p ocedu e can be desc ibed in he ollowing s eps. Fo ixed B
and o b= 1, ..., B,
1. Fo j= 1, . . . , k, le {ε∗
ij,b, i = 1, . . . , nj}be an i.i.d. sample om he dis i-
bu ion o (1 −a2
j)1/2Vj+ajZ, whe e Vjhas dis ibu ion ˜
Fεjand Zis, e.g.,
a s anda d no mal andom a iable. The cons an s aj, which de e mine he
amoun o smoo hing in he boo s ap, a e ela ed o he sample size in each
sample.
2. Fo j= 1, . . . , k, de ine new esponses unde he null hypo hesis Y∗
ij,b by
(i= 1, . . . , nj)
Y∗
ij,b = ˆm(Xij) + ˆσj(Xij)ε∗
ij,b.
3. Le T1∗
KS,b,T1∗
CM,b,T2∗
KS,b and T2∗
CM,b be he es s a is ics ob ained om he
boo s ap samples {(Xij, Y ∗
ij,b), i = 1, . . . , nj},j= 1, . . . , k.
Since in s ep 2 he boo s ap esamples a e cons uc ed unde he null hypo-
hesis o equal eg ession unc ions, his mechanism app oxima es he dis ibu ion
o he es s a is ics unde he null hypo hesis. I we deno e T1∗
KS,(b) o he o de
s a is ics o he alues T1∗
KS,1, . . . , T1∗
KS,B ob ained in s ep 3, and analogously o
T1∗
CM,(b),T2∗
KS,(b)and T2∗
CM,(b), hen T1∗
KS,([(1−α)B]),T1∗
CM,([(1−α)B]),T2∗
KS,([(1−α)B]) and
T2∗
CM,([(1−α)B]) app oxima e he (1 −α)-quan iles o he dis ibu ion o T1
KS,T1
CM ,
T2
KS and T2
CM unde he null hypo hesis espec i ely.
Tes ing o he equali y o k eg ession cu es 25
2.5 Simula ion s udy
This sec ion is de o ed o s udying he p ac ical beha io o he boo s ap p o-
cedu e by means o simula ions. In he i s pa we es ic ou s udy o he
compa ison o wo eg ession cu es in o de o be able o compa e ou me hod
wi h o he s in he li e a u e. Pa icula ly, we will compa e ou me hod wi h he
p ocedu e de eloped by Neumeye and De e (2003), which is based on a ma ked
empi ical p ocess app oach. The compa ison is ca ied ou o a selec ion o he
models conside ed in he simula ion sec ion o Neumeye and De e’s pape (excep
o model (ii), which is no conside ed in hei pape ). We conside he ollowing
models:
(i)m1(x) = 1; m2(x) = 1
(ii)m1(x) = x;m2(x) = x
(iii)m1(x) = sin(2πx); m2(x) = sin(2πx)
(i )m1(x) = exp(x); m2(x) = exp(x)
( )m1(x) = 1; m2(x) = 1 + x
( i)m1(x) = exp(x); m2(x) = exp(x) + x
( ii)m1(x) = sin(2πx); m2(x) = sin(2πx) + x
( iii)m1(x) = 1; m2(x) = 1 + sin(2πx)
Clea ly, models (i)−(i ) co espond o he null hypo hesis and models ( )−
( iii) o he al e na i e hypo hesis. In each case, we conside a homoscedas ic and
a he e oscedas ic si ua ion. In he homoscedas ic case he a iance unc ions a e
(as in Neumeye and De e, 2003)
σ2
1(x) = 0.25 and σ2
2(x) = 0.50,(2.10)
and in he he e oscedas ic case he a iance unc ions a e
σ2
1(x) = σ2
2(x) = ex
R1
0e d .(2.11)
The dis ibu ion o ε1and ε2is he s anda d no mal dis ibu ion. O he simu-
la ions ha e been ca ied ou wi h o he dis ibu ions o he e o s and simila
26 Chap e 2
esul s we e ob ained. In all cases he co a ia es X1and X2ha e uni o m dis i-
bu ion on [0,1].
Fo he ke nel unc ion needed in he es ima ion o he eg ession and a iance
cu es, we choose he Epanechniko ke nel K(u) = 0.75(1 −u2)I(|u|<1).
In he heo e ical esul s we wo k wi h only one bandwid h hnand we ha e
ound in simula ions a be e app oxima ion o he le el when he same bandwid h
is used o es ima e he common eg ession cu e and he eg ession cu es in each
popula ion, especially o ‘oscilla ing’ unc ions, as in model (iii). This can be ex-
plained as ollows: when using di e en bandwid hs he es ima ion o he eg ession
cu e in one popula ion can be o e smoo hed wi h espec o he es ima ion o he
common eg ession unc ion, and hen he es could de ec di e en cu es when
hey a e eally he same. Mo e p ecisely, we conside a bandwid h o he o m
h=cn−3/10, which e i ies he egula i y condi ions gi en in Sec ion 2.2. Cases
c= 0.5 and c= 1 a e p esen ed. In o he si ua ions he alue o cmus be adap ed
o he suppo o he eg esso a iables.
Conce ning he amoun o smoo hing we apply in he boo s ap, we ecommend
di e en cons an s ajdepending on he sample sizes nj. We wo k in all cases wi h
aj= 2n−3/10
j. When i is conside ed as a bandwid h, ajchosen in his way is a
small bandwid h o es ima e he densi y o he e o s (a s anda d no mal in ou
simula ions).
Tables 2.1-2.4 egis e he p opo ion o ejec ions in 1000 ials o sample sizes
(n1, n2) = (50,50),(100,50) and (100,100) and B= 200 boo s ap eplica ions.
The signi icance le els a e α= 0.05 and α= 0.10.
Tables 2.1 and 2.3 show ha he le el is well app oxima ed in mos cases and
o bo h bandwid h choices. The app oxima ion is be e o he es s based on he
s a is ics T1
KS and T1
CM . The es s based on T2
KS and T2
CM seem o be somewha
conse a i e. The beha io o he powe (Tables 2.2 and 2.4) o he es s based
on T1
KS and T1
CM is good o models ( ), ( i) and ( ii). On he o he hand, he
es s based on T2
KS and T2
CM gi e good powe o model ( iii). In bo h cases he
ob ained powe is be e o he la ge sample sizes. The C am´e - on Mises es
gi es be e powe han he Kolmogo o -Smi no es in mos si ua ions. Also no e

Tes ing o he equali y o k eg ession cu es 27
ha he choice o he bandwid h has li le impac on he ejec ion p obabili ies.
In Neumeye and De e (2003) only he homoscedas ic case is conside ed. Fo
a compa ison wi h hei simula ions we ound ha ou p ocedu e based on T1
KS
and T1
CM yields be e o simila esul s o he powe in mos cases o models
( ), ( i) and ( ii), whe eas o model ( iii) we ob ained be e esul s wi h he
es s based on T2
KS and T2
CM .
Ou me hod is alid o mo e han wo cu es. We explo e now he beha io o
he es ing p ocedu e in a h ee eg ession cu es se up. We conside he ollowing
models
(ix)m1(x) = x;m2(x) = x;m3(x) = x
(x)m1(x) = x;m2(x) = x+ 0.25; m3(x) = x+ 0.5
(xi)m1(x) = x;m2(x) = 0.5; m3(x) = 1 −x
Model (ix) co esponds o he null hypo hesis and models (x) and (xi) co espond
o he al e na i e. The a iance unc ions a e
σ2
1(x) = σ2
2(x) = σ2
3(x) = 0.5.(2.12)
As in he p e ious simula ed models, he co a ia es a e uni o mly dis ibu ed in
[0,1] and he e o s a e dis ibu ed as a s anda d no mal. The choice o he ke nel,
he bandwid h and he amoun o smoo hing in he boo s ap is he same as in he
p e ious simula ions: Kis he Epanechniko ke nel, h=cn−3/10 (cases c= 0.5
and c= 1 a e displayed) and aj= 2n−3/10
j. The ob ained esul s o bo h e sions
o he es s a is ics, wi h samples sizes (50,50,50), (100,50,50), (100,100,50)
and (100,100,100) and signi icance le els α= 0.05 and α= 0.10 a e shown in
Table 2.5.
The es s based on T1
KS and T1
CM app oxima e he le el well, while he es s
based on T2
KS and T2
CM seem o be a bi conse a i e, as happened in he wo-cu e
examples. In all cases he powe inc eases wi h he sample sizes. The i s e sion
o he es s a is ics p oduces be e powe in model (x) and he second e sion
gi es be e powe in model (xi).
28 Chap e 2
Table 2.1: Rejec ion p obabili ies unde he null hypo hesis –models (i) o (i )– o
he es s based on T1
KS and T1
CM . The models a e homoscedas ic, wi h a iances
gi en in (2.10), and he e oscedas ic, wi h a iances gi en in (2.11).
c= 0.5c= 1
T1
KS T1
CM T1
KS T1
CM
(n1, n2)α: 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
Homoscedas ic models
(50,50) (i) 0.048 0.095 0.050 0.108 0.051 0.093 0.052 0.105
(ii) 0.053 0.101 0.053 0.106 0.051 0.101 0.054 0.102
(iii) 0.066 0.102 0.058 0.107 0.064 0.110 0.071 0.122
(i ) 0.058 0.110 0.057 0.105 0.057 0.102 0.055 0.108
(100,50) (i) 0.063 0.111 0.057 0.103 0.055 0.099 0.055 0.107
(ii) 0.055 0.103 0.061 0.108 0.048 0.100 0.050 0.106
(iii) 0.060 0.111 0.065 0.118 0.067 0.117 0.076 0.127
(i ) 0.056 0.100 0.062 0.110 0.048 0.097 0.050 0.111
(100,100) (i) 0.054 0.100 0.053 0.100 0.052 0.088 0.050 0.106
(ii) 0.056 0.097 0.051 0.099 0.054 0.088 0.053 0.107
(iii) 0.049 0.094 0.050 0.102 0.058 0.110 0.061 0.110
(i ) 0.055 0.096 0.053 0.099 0.052 0.095 0.059 0.104
He e oscedas ic models
(50,50) (i) 0.050 0.090 0.050 0.098 0.052 0.097 0.050 0.095
(ii) 0.053 0.095 0.052 0.100 0.047 0.095 0.053 0.103
(iii) 0.055 0.106 0.060 0.105 0.049 0.096 0.052 0.103
(i ) 0.054 0.098 0.053 0.100 0.042 0.090 0.054 0.094
(100,50) (i) 0.059 0.102 0.054 0.104 0.054 0.099 0.052 0.114
(ii) 0.050 0.105 0.053 0.103 0.048 0.096 0.053 0.102
(iii) 0.058 0.109 0.060 0.099 0.060 0.116 0.063 0.121
(i ) 0.049 0.102 0.055 0.099 0.050 0.099 0.053 0.102
(100,100) (i) 0.055 0.097 0.049 0.102 0.050 0.096 0.056 0.099
(ii) 0.057 0.098 0.049 0.101 0.052 0.097 0.056 0.100
(iii) 0.048 0.094 0.046 0.098 0.054 0.098 0.056 0.107
(i ) 0.055 0.096 0.048 0.102 0.052 0.093 0.056 0.104
Tes ing o he equali y o k eg ession cu es 29
Table 2.2: Rejec ion p obabili ies unde he al e na i e hypo hesis –models ( ) o
( iii)– o he es s based on T1
KS and T1
CM . The models a e homoscedas ic, wi h
a iances gi en in (2.10), and he e oscedas ic, wi h a iances gi en in (2.11).
c= 0.5c= 1
T1
KS T1
CM T1
KS T1
CM
(n1, n2)α: 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
Homoscedas ic models
(50,50) ( ) 0.939 0.972 0.960 0.986 0.948 0.978 0.972 0.986
( i) 0.943 0.970 0.965 0.983 0.950 0.973 0.969 0.986
( ii) 0.950 0.966 0.966 0.977 0.945 0.972 0.963 0.977
( iii) 0.245 0.431 0.204 0.398 0.213 0.370 0.158 0.312
(100,50) ( ) 0.983 0.995 0.992 0.997 0.983 0.994 0.992 0.998
( i) 0.988 0.994 0.993 0.998 0.983 0.990 0.991 0.995
( ii) 0.986 0.993 0.990 0.998 0.977 0.986 0.984 0.994
( iii) 0.315 0.480 0.239 0.431 0.286 0.436 0.180 0.342
(100,100) ( ) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
( i) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
( ii) 1.000 1.000 1.000 1.000 0.999 1.000 1.000 1.000
( iii) 0.488 0.702 0.428 0.688 0.430 0.647 0.324 0.562
He e oscedas ic models
(50,50) ( ) 0.537 0.673 0.613 0.725 0.596 0.717 0.638 0.762
( i) 0.543 0.670 0.613 0.727 0.583 0.712 0.643 0.753
( ii) 0.564 0.666 0.612 0.719 0.586 0.701 0.626 0.748
( iii) 0.118 0.201 0.100 0.185 0.122 0.199 0.086 0.167
(100,50) ( ) 0.725 0.821 0.791 0.863 0.760 0.854 0.816 0.886
( i) 0.728 0.817 0.796 0.867 0.740 0.847 0.815 0.881
( ii) 0.727 0.820 0.793 0.862 0.749 0.844 0.810 0.870
( iii) 0.164 0.269 0.133 0.248 0.178 0.302 0.136 0.240
(100,100) ( ) 0.898 0.945 0.917 0.954 0.899 0.952 0.928 0.962
( i) 0.890 0.943 0.915 0.954 0.912 0.953 0.929 0.962
( ii) 0.888 0.939 0.920 0.953 0.890 0.951 0.920 0.958
( iii) 0.208 0.331 0.151 0.282 0.213 0.341 0.136 0.261
30 Chap e 2
Table 2.3: Rejec ion p obabili ies unde he null hypo hesis –models (i) o (i )– o
he es s based on T2
KS and T2
CM . The models a e homoscedas ic, wi h a iances
gi en in (2.10), and he e oscedas ic, wi h a iances gi en in (2.11).
c= 0.5c= 1
T2
KS T2
CM T2
KS T2
CM
(n1, n2)α: 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
Homoscedas ic models
(50,50) (i) 0.038 0.080 0.041 0.091 0.056 0.104 0.049 0.101
(ii) 0.047 0.084 0.045 0.101 0.049 0.095 0.055 0.100
(iii) 0.052 0.097 0.049 0.098 0.047 0.097 0.066 0.121
(i ) 0.052 0.091 0.050 0.100 0.048 0.092 0.055 0.096
(100,50) (i) 0.046 0.092 0.045 0.081 0.048 0.088 0.050 0.105
(ii) 0.043 0.080 0.048 0.078 0.054 0.093 0.055 0.101
(iii) 0.046 0.096 0.048 0.101 0.067 0.130 0.073 0.130
(i ) 0.045 0.092 0.042 0.084 0.059 0.099 0.055 0.110
(100,100) (i) 0.030 0.081 0.034 0.062 0.039 0.090 0.044 0.087
(ii) 0.033 0.073 0.033 0.064 0.044 0.077 0.044 0.092
(iii) 0.044 0.089 0.046 0.080 0.045 0.091 0.060 0.104
(i ) 0.039 0.070 0.032 0.065 0.044 0.086 0.049 0.095
He e oscedas ic models
(50,50) (i) 0.046 0.092 0.043 0.087 0.044 0.095 0.057 0.094
(ii) 0.047 0.084 0.043 0.089 0.043 0.095 0.051 0.088
(iii) 0.050 0.087 0.045 0.080 0.044 0.088 0.047 0.087
(i ) 0.040 0.082 0.044 0.091 0.038 0.086 0.046 0.089
(100,50) (i) 0.040 0.073 0.043 0.075 0.042 0.085 0.049 0.087
(ii) 0.036 0.075 0.039 0.076 0.032 0.083 0.038 0.085
(iii) 0.044 0.080 0.036 0.083 0.055 0.101 0.041 0.090
(i ) 0.036 0.080 0.039 0.079 0.039 0.079 0.046 0.082
(100,100) (i) 0.041 0.079 0.034 0.057 0.039 0.067 0.043 0.076
(ii) 0.042 0.072 0.030 0.060 0.034 0.079 0.039 0.073
(iii) 0.038 0.075 0.033 0.063 0.038 0.093 0.051 0.101
(i ) 0.035 0.066 0.032 0.063 0.038 0.083 0.044 0.074
Tes ing o he equali y o k eg ession cu es 37
We conside he ollowing es s a is ics: a Kolmogo o -Smi no ype s a is ic
SKS =
k
X
j=1
sup
y|ˆ
Uj(y)|,
and a C am´e - on Mises ype s a is ic
SCM =
k
X
j=1 Zˆ
U2
j(y)dˆ
Fε(y).
The null hypo hesis (2.13) is ejec ed when he ob ained alue o he es s a is ic
SKS o SCM is la ge han a ce ain c i ical alue.
2.7.1 Asymp o ic esul s
In he ollowing heo em and co olla y we s a e he weak con e gence o he p ocess
ˆ
U(y) and gi e he asymp o ic dis ibu ions o he es s a is ics SKS and SCM .
The egula i y assump ions a e basically he same we lis ed in he beginning o
Sec ion 2.3, excep ha now he co a ia es do no need o ha e common suppo
(assump ion A1-i). The p oo s a e included in Sec ion 2.8.
Theo em 2.5 Assume (A1)-(A4). Then, unde he null hypo hesis H0ε, he k-
dimensional p ocess ˆ
U(y) = ( ˆ
U1(y), . . . , ˆ
Uk(y)) con e ges weakly o a cen e ed
k-dimensional Gaussian p ocess U(y) = (U1(y), . . . , Uk(y)) wi h co a iance s uc-
u e gi en by
Co (Uj(y), Uj0(y0))
=
k
X
l=1
p1/2
jp1/2
j0µ1−I(l=j)
pj¶µ1−I(l=j0)
pj0¶E(ϕl(Xl, Yl, y)ϕl(Xl, Yl, y0)),
whe e, o j= 1, . . . , k,
ϕj(u, , y) =Iµ −mj(u)
σj(u)≤y¶−Fε(y)
+y ε(y)( −mj(u))2−σ2
j(u)
2σ2
j(u)+ ε(y) −mj(u)
σj(u).

38 Chap e 2
Co olla y 2.3 Assume (A1)-(A4). Then, unde he null hypo hesis H0ε,
SKS
d
→
k
X
j=1
sup |Uj(y)|,
SCM
d
→
k
X
j=1 ZU2
j(y)dFε(y).
2.7.2 Boo s ap, simula ions and da a analysis
Boo s ap app oxima ion. In p ac ical applica ions, he c i ical alues o he
es s a is ics SKS and SCM can be app oxima ed by he boo s ap p ocedu e
desc ibed below. Le ˜
Fεbe he s anda dized e sion o he empi ical dis ibu ion
unc ion o he join sample o es ima ed esiduals conside ed in (2.14).
Fo ixed Band o b= 1, ..., B,
1. Fo j= 1, . . . , k, le {ε∗
ij,b, i = 1, . . . , nj}be an i.i.d. sample om he dis i-
bu ion o (1 −a2
j)1/2V+ajZ, whe e Vhas dis ibu ion ˜
Fεand Zis, e.g., a
s anda d no mal andom a iable.
2. Fo j= 1, . . . , k, and i= 1, . . . , nj, de ine new esponses unde he null
hypo hesis Y∗
ij,b by
Y∗
ij,b = ˆmj(Xij) + ˆσj(Xij)ε∗
ij,b
3. Le S∗
KS,b and S∗
CM,b be he es s a is ics ob ained om he boo s ap sam-
ples {(Xij, Y ∗
ij,b), i = 1, . . . , nj},j= 1, . . . , k.
In s ep 2 he boo s ap esamples a e cons uc ed unde he null hypo hesis
o equal dis ibu ion o he e o s since we d aw esiduals om he combined
sample o es ima ed esiduals in all popula ions. This mechanism app oxima es
he dis ibu ion o he es s a is ics unde he null hypo hesis. I we deno e
S∗
KS,(b)( espec i ely, S∗
CM,(b)) o he o de s a is ics o he alues S∗
KS,1, . . . , S∗
KS,B
ob ained in s ep 3 ( espec i ely, S∗
CM,1, . . . , S∗
CM,B) hen S∗
KS,([(1−α)B]) ( espec i ely,
S∗
CM,([(1−α)B])) app oxima es he (1 −α)-quan iles o he dis ibu ion o SKS ( es-
pec i ely, SCM ) unde he null hypo hesis.
Tes ing o he equali y o k eg ession cu es 39
Table 2.6: Rejec ion p obabili ies unde models (i) o (iii) o he es s based on
SKS and SCM .
c= 0.5c= 1
SKS SCM SKS SCM
(n1, n2)α: 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
(50,50) (i) 0.061 0.092 0.067 0.115 0.050 0.113 0.053 0.123
(ii) 0.613 0.727 0.775 0.841 0.627 0.747 0.794 0.859
(iii) 0.088 0.165 0.143 0.239 0.106 0.183 0.179 0.290
(100,50) (i) 0.059 0.105 0.060 0.101 0.057 0.100 0.052 0.102
(ii) 0.721 0.812 0.837 0.906 0.733 0.832 0.870 0.920
(iii) 0.146 0.231 0.218 0.321 0.147 0.242 0.230 0.350
(100,100) (i) 0.054 0.096 0.062 0.107 0.061 0.103 0.056 0.107
(ii) 0.919 0.961 0.977 0.989 0.931 0.964 0.983 0.993
(iii) 0.225 0.353 0.377 0.495 0.265 0.401 0.446 0.579
Simula ions. We show a simula ed example wi h wo popula ions. The mod-
els o he dis ibu ions o he e o s a e
(i)ε1, ε2∼N(0,1)
(ii)ε1∼N(0,1), ε2∼Exponen ial(1) −1
(iii)ε1∼N(0,1), ε2∼Uni o m[−1/√12,1/√12]
Model (i) co esponds o he null hypo hesis, and models (ii) and (iii) co es-
pond o he al e na i e hypo hesis. In all cases he co a ia es ollow a uni o m
dis ibu ion on [0,1]. The esponses a e gi en by he eg ession unc ions m1(x) =
m2(x) = xand he a iance unc ions σ1(x) = σ2(x) = 0.50.
In his case we es ima e e e y needed cu e o cons uc he esiduals o m each
sample sepa a ely. F om a p ac ical poin o iew, we ecommend o choose di e-
en bandwid hs acco ding o he sample sizes in each popula ion. Mo e p ecisely,
we conside bandwid hs hj=cn−3/10
j o es ima e mjand σjin each popula ion.
Cases c= 0.5 and c= 1 a e displayed. The amoun o smoo hing in he boo s ap
was chosen o be aj= 2n−3/10
j. We wo k wi h he Epanechniko ke nel K(u) =
40 Chap e 2
0.75(1 −u2)I(|u|<1).
Table 2.6 shows he p opo ion o ejec ions in 1000 ials o sample sizes
(n1, n2) = (50,50),(100,50) and (100,100) and B= 200 boo s ap eplica ions.
The signi icance le els a e α= 0.05 and α= 0.10.
The le el is well app oxima ed o bo h bandwid h choices and he powe in-
c eases wi h he sample sizes. The C am´e - on Mises es gi es be e powe han
he Kolmogo o -Smi no es in mos si ua ions. As in he compa ison o eg es-
sion cu es p oblem, he choice o he bandwid h has li le impac on he ejec ion
p obabili ies.
Real da a analysis. We ha e also applied his es ing p ocedu e o he da a
desc ibed in Sec ion 2.6. We ha e es ed he null hypo hesis o equal dis ibu ion
o he esiduals in he h ee popula ions. The esul s con i m his hypo hesis.
The es was ca ied ou o di e en alues o c anging om 0.5 o 1.5. The
p- alues we e calcula ed om 1000 boo s ap eplica ions. Fo he Kolmogo o -
Smi no ype s a is ic SKS he p- alues we e be ween 0.33 and 0.65 and o he
C am´e - on Mises ype s a is ic SCM he p- alues we e be ween 0.21 and 0.36.
2.8 P oo s
P oo o Theo em 2.1. 1s pa . Assume Fεj0(y) = Fεj(y). This implies ha
he i s and he second momen o hese dis ibu ions a e he same. F om he
i s momen we ha e ha
EµYj−m(Xj)
σj(Xj)¶=EµYj−mj(Xj)
σj(Xj)¶= 0.
The second momen o Fεj0(y) can be w i en as
V a µYj−m(Xj)
σj(Xj)¶
=E"µYj−m(Xj)
σj(Xj)¶2#=E"µYj−mj(Xj) + mj(Xj)−m(Xj)
σj(Xj)¶2#
Tes ing o he equali y o k eg ession cu es 41
=V a µYj−mj(Xj)
σj(Xj)¶+ 2Eµ(Yj−mj(Xj))(mj(Xj)−m(Xj))
σ2
j(Xj)¶
+Eµ(mj(Xj)−m(Xj))2
σ2
j(Xj)¶.
We a e assuming ha
V a µYj−m(Xj)
σj(Xj)¶=V a µYj−mj(Xj)
σj(Xj)¶,
and clea ly
2Eµ(Yj−mj(Xj))(mj(Xj)−m(Xj))
σ2
j(Xj)¶= 0.
Hence
Eµ(mj(Xj)−m(Xj))2
σ2
j(Xj)¶= 0,
and his implies mj(x) = m(x), o all j= 1, . . . , k, and o all x∈RX, excep
o a se o poin s o p obabili y ze o. The con inui y o he unc ions mjallows
ex ending he equali y o all x∈RX.
2nd pa . Assume Fε0(y) = Fε(y). As in he p e ious pa , i we compu e he i s
momen we ob ain
k
X
j=1
pjEµYj−m(Xj)
σj(Xj)¶=
k
X
j=1
pjEµYj−mj(Xj)
σj(Xj)¶= 0.
And om he second momen
V a Ãk
X
j=1
pj
Yj−m(Xj)
σj(Xj)!=
k
X
j=1
p2
jE"µYj−m(Xj)
σj(Xj)¶2#
=
k
X
j=1
p2
jV a µYj−mj(Xj)
σj(Xj)¶+
k
X
j=1
p2
jEµ(mj(Xj)−m(Xj))2
σ2
j(Xj)¶
=V a Ãk
X
j=1
pj
Yj−mj(Xj)
σj(Xj)!+
k
X
j=1
p2
jEµ(mj(Xj)−m(Xj))2
σ2
j(Xj)¶.
Then i holds ha
k
X
j=1
p2
jEµ(mj(Xj)−m(Xj))2
σ2
j(Xj)¶= 0.
42 Chap e 2
This and he con inui y o he unc ions mjimplies mj(x) = m(x) o all j=
1, . . . , k and o all x∈RX.
The con e se implica ions a e i ial.
Be o e p o ing he main esul s in Sec ions 2.3 and 2.7, we s a e he ollowing
h ee auxilia y lemmas.
Lemma 2.1 Assume (A1)-(A3). Then, unde he null hypo hesis H0, o any
j= 1, . . . , k,
Zˆm(x)−m(x)
σj(x) Xj(x)dx =1
n
k
X
l=1
nl
X
i=1
Yil −m(Xil)
σj(Xil)
Xj(Xil)
mix(Xil)+oP(n−1/2).
P oo . Conside he ke nel es ima o o he densi y o he mix u e mix(x)
ˆ
mix(x) = 1
nhn
k
X
l=1
nl
X
i=1
Kµx−Xil
hn¶.
I is well known ha (see, e.g., Wand and Jones, 1995) ˆ
mix(x) = mix(x)+O(h2
n)+
OP((nhn)−1/2), uni o mly in x. By assump ion (A2-ii) we ha e ha O(h2
n) =
O((nhn)−1/2), hence ˆ
mix(x)− mix(x) = OP((nhn)−1/2) and ˆ
mix(x) −1
mix(x)−1 =
OP((nhn)−1/2). Simila ly ˆm(x)−m(x) = OP((nhn)−1/2). Then we ob ain
ˆm(x)−m(x) = 1
nhnˆ
mix(x)
k
X
l=1
nl
X
i=1
Kµx−Xil
hn¶(Yil −m(x))
=1
nhn mix(x)
k
X
l=1
nl
X
i=1
Kµx−Xil
hn¶(Yil −m(x)) + OP((nhn)−1)),
uni o mly in x. Taking his and condi ion (A2-ii) in o accoun , he in eg al be-
comes
Zˆm(x)−m(x)
σj(x) Xj(x)dx
=1
nhn
k
X
l=1
nl
X
i=1 ZK((x−Xil)h−1
n)(Yil −m(x))
σj(x)
Xj(x)
mix(x)dx +oP(n−1/2).

Tes ing o he equali y o k eg ession cu es 43
Deno e L(x) = (Yil −m(x)) Xj(x)( mix(x)σj(x))−1. Using he change o a i-
able u= (x−Xil)h−1
n, a Taylo expansion o second o de o La ound Xil and
assump ion (A3), we ha e ha
ZK((x−Xil)h−1
n)(Yil −m(x))
σj(x)
Xj(x)
mix(x)dx
=hnZK(u)L(Xil +hnu)du
=hnL(Xil)ZK(u)du +h2
nL0(Xil)ZuK(u)du +OP(h3
n)
=hnL(Xil) + OP(h3
n).
Then, using assump ion (A2),
Zˆm(x)−m(x)
σj(x) Xj(x)dx =1
n
k
X
l=1
nl
X
i=1
Yil −m(Xil)
σj(Xil)
Xj(Xil)
mix(Xil)+oP(n−1/2).
Lemma 2.2 Assume (A1)-(A3). Then, o any j= 1, . . . , k,
Zˆmj(x)−mj(x)
σj(x) Xj(x)dx =1
nj
nj
X
i=1
Yij −mj(Xij)
σj(Xij)+oP(n−1/2
j).
P oo . Simila o he p oo o he p e ious lemma.
Lemma 2.3 Assume (A1)-(A3). Then, o any j= 1, . . . , k
Zˆσj(x)−σj(x)
σj(x) Xj(x)dx =1
nj
nj
X
i=1
(Yij −mj(Xij)2−σ2
j(Xij)
2σ2
j(Xij)+oP(n−1/2
j).
P oo . W i e
ˆσj(x)−σj(x) = ˆσ2
j(x)−σ2
j(x)
2σj(x)−(ˆσj(x)−σj(x))2
2σj(x).
F om P oposi ion 3 in Ak i as and Van Keilegom (2001) he second e m is o
o de OP((njhn)−1), and hence we can w i e
ˆσj(x)−σj(x) = ˆσ2
j(x)−σ2
j(x)
2σj(x)+OP((njhn)−1)
44 Chap e 2
uni o mly in x, and i we deno e ˜σ2
j(x) = Pnj
i=1 W(j)
ij (x, hn)(Yij −mj(x))2,
ˆσ2
j(x) =
nj
X
i=1
W(j)
ij (x, hn)Y2
ij −ˆm2
j(x)
=
nj
X
i=1
W(j)
ij (x, hn)(Y2
ij −ˆm2
j(x))
=
nj
X
i=1
W(j)
ij (x, hn)(Yij −mj(x))2−(mj(x)−ˆmj(x))2
= ˜σ2
j(x) + OP((njhn)−1),
because ˆmj(x)−mj(x) = OP((njhn)−1/2). Since σ2
j(x) can be conside ed as he
eg ession unc ion o he a iable (Yij −mj(x))2, we ha e ha ˜σ2
j(x)−σ2
j(x) =
OP((njhn)−1/2). Now we can w i e
˜σ2
j(x)−σ2
j(x) = 1
njhnˆ
Xj(x)
nj
X
i=1
Kµx−Xij
hn¶((Yij −mj(x))2−σ2
j(x))
=1
njhn Xj(x)
nj
X
i=1
Kµx−Xij
hn¶((Yij −mj(x))2−σ2
j(x))
+Ã1−
ˆ
Xj(x)
Xj(x)!(˜σ2
j(x)−σ2
j(x))
=1
njhn Xj(x)
nj
X
i=1
Kµx−Xij
hn¶((Yij −mj(x))2−σ2
j(x))
+OP((njhn)−1).
Using (A2-ii) w i e
Zˆσj(x)−σj(x)
σj(x) Xj(x)dx
=Z˜σ2
j(x)−σ2
j(x)
2σ2
j(x) Xj(x)dx +OP((njhn)−1)
=1
njhn
nj
X
i=1 ZK((x−Xij)h−1
n)((Yij −mj(x))2−σ2
j(x))
2σ2
j(x)dx +oP(n−1/2
j).
Tes ing o he equali y o k eg ession cu es 45
Using a Taylo expansion o second o de we ob ain
Zˆσj(x)−σj(x)
σj(x) Xj(x)dx
=1
nj
nj
X
i=1
(Yij −mj(Xij))2−σ2
j(Xij)
2σ2
j(Xij)+OP(h2
n) + oP(n−1/2
j)
=1
nj
nj
X
i=1
(Yij −mj(Xij))2−σ2
j(Xij)
2σ2
j(Xij)+oP(n−1/2
j),
and his concludes he p oo o he lemma.
P oo o Theo em 2.2. W i e
ˆ
Fεj0(y)−ˆ
Fεj(y) = ( ˆ
Fεj0(y)−Fεj(y)) −(ˆ
Fεj(y)−Fεj(y)).(2.15)
Fi s we will s udy he asymp o ic beha io o ˆ
Fεj0(y)−Fεj(y). We will use some
esul s and p oo s om Ak i as and Van Keilegom (2001). These au ho s assume
ha he unc ions m,mj, and σja e L- unc ionals depending on a ce ain sco e
unc ion J. In ou case hese unc ionals a e he condi ional mean and a iance,
ha co espond o J≡1. This choice o Jis no co e ed by he esul s in Ak i as
and Van Keilegom (2001). Howe e , i is easy o check ha he esul s in ha
pape can be ex ended o J≡1. F om he p oo o Theo em 1 in Ak i as and Van
Keilegom (2001) we ha e ha
ˆ
Fεj0(y)−Fεj(y) = 1
nj
nj
X
i=1
IµYij −m(Xij)
σj(Xij)≤y¶−Fεj(y) (2.16)
+ εj(y)Zy(ˆσj(x)−σj(x)) + ˆm(x)−m(x)
σj(x) Xj(x)dx +Rnj(y),
whe e supy|Rnj(y)|=oP(n−1/2
j). No e ha in Ak i as and Van Keilegom (2001)
he es ima ion o he dis ibu ion o he esiduals is conside ed om one sam-
ple. This means ha , wi h ou no a ion, he e o εij is es ima ed by (Yij −
ˆmj(Xij))/ˆσj(Xij). Howe e , he decomposi ion gi en in (2.16) emains alid when
he e o s a e es ima ed wi h ˆm, because hei Lemma 1 holds in ha case.
46 Chap e 2
The in eg al on he igh side abo e can be decomposed as ollows:
Zy(ˆσj(x)−σj(x)) + ˆm(x)−m(x)
σj(x) Xj(x)dx
=yZˆσj(x)−σj(x)
σj(x) Xj(x)dx +Zˆm(x)−m(x)
σj(x) Xj(x)dx.
Now, using Lemma 2.1, Lemma 2.3 and he ac ha unde he null hypo hesis
m=m1=··· =mk,
ˆ
Fεj0(y)−Fεj(y) = 1
nj
nj
X
i=1
IµYij −m(Xij)
σj(Xij)≤y¶−Fεj(y) (2.17)
+y εj(y)1
nj
nj
X
i=1
(Yij −m(Xij))2−σ2
j(Xij)
2σ2
j(Xij)
+ εj(y)1
n
k
X
l=1
nl
X
i=1
Yil −m(Xil)
σj(Xil)
Xj(Xil)
mix(Xil)+oP(n−1/2)
uni o mly in y. No e ha nj=O(n) because o condi ion (A2-i).
Again om he p oo o Theo em 1 in Ak i as and Van Keilegom (2001)
ˆ
Fεj(y)−Fεj(y) = 1
nj
nj
X
i=1
IµYij −m(Xij)
σj(Xij)≤y¶−Fεj(y)
+ εj(y)Zy(ˆσj(x)−σj(x)) + ˆmj(x)−m(x)
σj(x) Xj(x)dx +oP(n−1/2
j),
and using Lemma 2.2 and Lemma 2.3
ˆ
Fεj(y)−Fεj(y) = 1
nj
nj
X
i=1
IµYij −m(Xij)
σj(Xij)≤y¶−Fεj(y) (2.18)
+y εj(y)1
nj
nj
X
i=1
(Yij −m(Xij))2−σ2
j(Xij)
2σ2
j(Xij)
+ εj(y)1
nj
nj
X
i=1
Yij −m(Xij)
σj(Xij)+oP(n−1/2
j)
uni o mly in y.
Tes ing o he equali y o k eg ession cu es 53
whe e dn2j(x) = 1 and dn1j(x) = (R(x)− j(x))σ−1
j(x). Clea ly, hese unc ions
a e in he condi ions o he p oo o Lemma 1 o he a o emen ioned pape . Using
hei no a ion dn2j∈˜
C1+δ
2(RX) o all δ > 0 and P(n−1/2dn1j∈C1+δ
1(RX)) →1
as n→ ∞ o 0 < δ ≤1, since dn1jis wice con inuously di e en iable on he
compac se RXby assump ion (A1).
Hence aking in o accoun (2.24) and (2.22), we can w i e
1
nj
nj
X
i=1
IµYij −mn(Xij)
σj(Xij)≤y¶(2.25)
=1
nj
nj
X
i=1
IµYij −mjn(Xij)
σj(Xij)−n−1/2R(Xij)− j(Xij)
σj(Xij)≤y¶
=1
nj
nj
X
i=1
IµYij −mjn(Xij)
σj(Xij)≤y¶+n−1/2 εj(y)E·R(Xj)− j(Xj)
σj(Xj)¸+oP(n−1/2).
As in he p oo o Theo em 2.3 we ha e
ˆ
Fεj(y)−Fεj(y) = 1
nj
nj
X
i=1
IµYij −mjn(Xij)
σj(Xij)≤y¶−Fεj(y) (2.26)
+ εj(y)Zy(ˆσj(x)−σj(x))
σj(x) Xj(x)dx
+ εj(y)Zˆmj(x)−mjn(x)
σj(x) Xj(x)dx +oP(n−1/2).
By combining (2.23), (2.25) and (2.26) we ob ain
ˆ
Fεj0(y)−ˆ
Fεj(y)
= εj(y)Zˆm(x)−mn(x)
σj(x) Xj(x)dx − εj(y)Zˆmj(x)−mjn(x)
σj(x) Xj(x)dx
+n−1/2 εj(y)E·R(Xj)− j(Xj)
σj(Xj)¸+oP(n−1/2),
and ollowing a simila p ocedu e as in Lemmas 2.2 and 2.3, we can w i e
ˆ
Fεj0(y)−ˆ
Fεj(y) (2.27)
= εj(y)
k
X
l=1
pl(1
nl
nl
X
i=1
Yil −mln(Xil)
σj(Xil)µ Xj(Xil)
mix(Xil)−I(l=j)
pj¶)
+n−1/2 εj(y)E·R(Xj)− j(Xj)
σj(Xj)¸+oP(n−1/2).

54 Chap e 2
No e ha (Yl−mln(Xl))/σj(Xil) = εl, which does no depend on n. The leading
e m o he ep esen a ion gi en in (2.27) unde Hl.a. is he same as he leading e m
o he ep esen a ion gi en in Theo em 2.3 unde he null hypo hesis. The e o e
hei limi dis ibu ions a e he same. This concludes he p oo .
P oo o Co olla y 2.2. Simila o he p oo o Co olla y 2.1, only by aking
in o accoun he weak con e gence o ˆ
W(y) unde Hl.a. gi en in Theo em 2.4.
P oo o Theo em 2.5. We use he ep esen a ion gi en in (2.18)
ˆ
Fεj(y)−Fε(y) = 1
nj
nj
X
i=1
ϕj(Xij, Yij, y) + oP(n−1/2
j),
uni o mly in y, whe e
ϕj(u, , y) =Iµ −mj(u)
σj(u)≤y¶−Fε(y)
+y ε(y)( −mj(u))2−σ2
j(u)
2σ2
j(u)+ ε(y) −mj(u)
σj(u).
No e ha
ˆ
Fε(y)−Fε(y) =
k
X
l=1
nl
n(ˆ
Fεl(y)−Fε(y)).
By simply w i ing ˆ
Fε(y)−ˆ
Fεj(y) = ( ˆ
Fε(y)−Fε(y)) −(ˆ
Fεj(y)−Fε(y)), we ha e
ˆ
Fε(y)−ˆ
Fεj(y) =
k
X
l=1 ³nl
n−I(l=j)´1
nl
nl
X
i=1
ϕl(Xil, Yil, y) + oP(n−1/2).
To p oo he weak con e gence o he k-dimensional p ocess we will make use
o he C am´e -Wold de ice, as we did in he p oo o Theo em 2.3. Le ˆ
V(y) =
Pn
j=1 ajˆ
Uj(y) be a linea combina ion o he componen s o he mul idimensional
p ocess. I su ices o show he weak con e gence o ˆ
V(y). I is easy o p o e ha
ˆ
V(y) =
k
X
j=1
(p1/2
jA−aj)1
n1/2
j
nj
X
i=1
ϕj(Xij, Yij, y) + oP(1),
Tes ing o he equali y o k eg ession cu es 55
whe e A=Pk
l=1 p1/2
lal. The leading e m o he p ocess ˆ
V(y) consis s o a sum
o kindependen p ocesses (mul iplied by a cons an , which will no in luence he
weak con e gence). Now w i e
ˆ
V(y) =
k
X
j=1
(p1/2
jA−aj)ˆ
Vj(y) + oP(1),
whe e ˆ
Vj(y) is he p ocess Gj-indexed by he class o unc ions
Gj={(u, )→ϕj(u, , y),−∞ < y < ∞}.
The class o unc ions Gjcan be w i en as
Gj=Gj1+Gj2+Gj3+Gj4,
whe e
Gj1=½(u, )→Iµ −mj(u)
σj(u)≤y¶,−∞ < y < ∞¾,
Gj2={(u, )→ −Fε(y),−∞ < y < ∞},
Gj3=½(u, )→y ε(y)( −mj(u))2−σ2
j(u)
2σ2
j(u),−∞ < y < ∞¾,
Gj4=½(u, )→ ε(y) −mj(u)
σj(u),−∞ < y < ∞¾.
The classes Gj3and Gj4 ac o ize in a pa no depending on yand a bounded
unc ion o y. Also he class Gj2can be ea ed in he same way, since i does no
depend on (u, ). Le Mbe such ha supy{|y ε(y)|,| ε(y)|,|Fε(y)|} < M. Hence,
o l= 2,3,4, N[ ](δ, Gjl, L2(P)) ≤2Mδ−1i δ < 2Mand N[ ](δ, Gjl, L2(P)) = 1 i
δ > 2M.
The unc ions in class Gj1a e dec easing in ( −mj(u))/σj(u). Then, by Theo-
em 2.7.5 in an de Vaa and Wellne (1996), N[ ](δ, Gj1, L2(P)) = O(exp(Kδ−1))
o some cons an K > 0. Ob iously, since he unc ions in Gj1a e bounded wi h
alues in [0,1], i δ > 1 hen N[ ](δ, Gj1, L2(P)) = 1.
These ea u es o he b acke ing numbe s o he classes in ol ed in he class
Gjensu e he weak con e gence o he p ocess ˆ
Vj(y), as we explained in he p oo
56 Chap e 2
o Theo em 2.3. The e o e we can conclude ha ou p ocess o in e es ˆ
V(y) is
weakly con e gen , and by he C am´e -Wold de ice so i is ˆ
U(y). The co a iance
s uc u e gi en in he s a emen o he Theo em can be ob ained immedia ely.
P oo o Co olla y 2.3. The p oo o he con e gence o he es s a is ics SKS
and SCM can be ob ained by ollowing he same a gumen s as in he p oo o
Co olla y 2.1.
Chap e 3
Compa ison o eg ession cu es
wi h censo ed esponses
In his chap e we conside he p oblem o compa ison o eg ession cu es when he
esponse a iables a e censo ed. The es is based on a compa ison o Kaplan-Meie
es ima o s o he dis ibu ion o he censo ed esiduals. As in he p e ious chap e ,
Kolmogo o -Smi no and C am´e - on Mises ype s a is ics a e conside ed. Some
asymp o ic esul s a e p o ed: weak con e gence o he p ocess o in e es , con-
e gence o he es s a is ics and beha io o he p ocess unde local al e na i es.
We also desc ibe a boo s ap p ocedu e in o de o app oxima e he c i ical alues
o he es . A simula ion s udy and an applica ion o a eal da a se conclude he
chap e .
This chap e is based on Pa do-Fe n´andez and Van Keilegom (2005) and i
ex ends he ideas and esul s ob ained in Chap e 2.
3.1 Mo i a ion and s a is ical model
Reg ession models a e used o desc ibing he ela ionship be ween a esponse and
a co a ia e. In he ield o su i al analysis i can be use ul o allow o censo ing
in he esponse a iable. Fo ins ance, we can conside a model whe e he su i al
ime ( o pa ien s ha ing a ce ain disease) is he esponse a iable and he age
57
58 Chap e 3
is he co a ia e. I we can dis inguish wo o mo e g oups in he popula ion (by
gende , ea ed pa ien s and non- ea ed pa ien s, e c.), we may be in e es ed in
es ing o he equali y o he co esponding eg ession cu es. This kind o es
allows us o check whe he he e ec o he co a ia e o e he a iable o in e es
is he same in all g oups.
As poin ed ou in Fan and Gijbels (1994), when he esponse a iable is cen-
so ed he usual ools o eg ession (sca e plo s, esiduals plo s, e c.) a e no
di ec ly applicable o check, a leas isually, he shape o he eg ession cu es.
This mo i a es he de elopmen o analy ic ools in censo ed eg ession.
In his con ex , he s a is ical model can be desc ibed as ollows. Le (Xj, Yj),
j= 1, . . . , k, be independen andom ec o s, whe e Yj ep esen s a ce ain es-
ponse a iable associa ed wi h he co a ia e Xj. Suppose ha he co a ia es ha e
common suppo RX. Assume ha , o j= 1, . . . , k, he esponse a iable Yj
is subjec o andom igh censo ing. This means ha he e exis s a censo ing
a iable Cj, independen o Yjgi en Xj, such ha we can obse e Zj= min{Yj, Cj}
and he indica o o censo ing ∆j=I(Yj≤Cj). Fo j= 1, . . . , k, assume ha he
ollowing non-pa ame ic eg ession models hold
Yj=mj(Xj) + σj(Xj)εj,
whe e he e o a iable εjis independen o Xj,mjis an unknown condi ional
loca ion unc ion
mj(x) = Z1
0
F−1
j(s|x)J(s)ds (3.1)
and σjis an unknown condi ional scale unc ion ep esen ing possible he e oscedas-
ici y
σ2
j(x) = Z1
0
F−1
j(s|x)2J(s)ds −m2
j(x),(3.2)
whe e Fj(·|x) is he condi ional dis ibu ion o Yjgi en he alue xo he co a ia e
Xj,F−1
j(s|x) = in { ;Fj( |x)≥s}is he co esponding quan ile unc ion and J(s)
is a sco e unc ion sa is ying R1
0J(s)ds = 1. Deno e Fεj(y) = P(εj≤y) o he
dis ibu ion o he e o εjin popula ion j. By he de ini ions (3.1) and (3.2),
R1
0F−1
εj(s)J(s)ds = 0 and R1
0F−1
εj(s)2J(s)ds = 1.

Compa ison o eg ession cu es wi h censo ed esponses 59
The choice o he sco e unc ion Jleads o di e en loca ion and scale unc ions.
In pa icula i J(s) = I(0 ≤s≤1) hen mj(x) = E(Yj|Xj=x) is he condi ional
mean unc ion and σ2
j(x) = V a (Yj|Xj=x) is he condi ional a iance unc ion.
Howe e , i may happen ha his choice o Jis no app op ia e because o he
inconsis ency o he es ima o o he condi ional dis ibu ion Fj(·|x) in he igh
ail due o he censo ing (see Sec ion 1.2). A use ul choice is
J(s) = 1
q−pI(p≤s≤q)
o some 0 ≤p < q ≤1, which leads o immed means and immed a iances.
The condi ional median o o he condi ional quan iles can be seen as limi s o
immed means.
The samples a e (Xij, Zij,∆ij), i= 1, ..., nj, om he dis ibu ion o (Xj, Zj,∆j),
o j= 1, . . . , k. Deno e n=Pk
j=1 nj.
We a e in e es ed in es ing he null hypo hesis o equali y be ween he loca ion
( eg ession) unc ions
H0:m1=m2=··· =mk,(3.3)
e sus he al e na i e
Ha:mi6=mj o some i, j ∈ {1, . . . , k}.
When he dis ibu ions o he e o s and he a iance unc ions a e he same
in all he g oups (we do no assume so, bu i is an in e es ing si ua ion), i he
null hypo hesis holds o a pa icula de ini ion o he loca ion unc ion, ha is o
a pa icula choice o J, hen i holds o all possible loca ion unc ions. Howe e ,
in a gene al si ua ion wi h di e en a iances o di e en e o dis ibu ions, H0
can be ue o a pa icula choice o he unc ions mjand alse o ano he one.
In Chap e 2 a mechanism o compa ison o eg ession cu es o comple e da a
was de eloped ia he es ima ion o he dis ibu ion o he e o s o he eg ession
models. The idea o he es ing p ocedu e is o compa e wo es ima o s o he
dis ibu ion o he e o s in each popula ion. In his chap e we will ex end ha
me hodology o he si ua ion whe e he esponse a iable may be censo ed. Now,
60 Chap e 3
because o he censo ing in he esponse a iable, we will conside
Zij −ˆmj(Xij)
ˆσj(Xij)
and
Zij −ˆm(Xij)
ˆσj(Xij)
o es ima e he censo ed esiduals, and we will subs i u e he empi ical dis ibu ion
by he Kaplan-Meie es ima o o he dis ibu ion unde andom censo ing (Kaplan
and Meie , 1958).
The chap e is o ganized as ollows. In Sec ion 3.2 we will explain he es ing
p ocedu e. In Sec ion 3.3 we will s a e he main asymp o ic esul s. A boo s ap
p ocedu e o app oxima e he c i ical poin s o he es is desc ibed in Sec ion 3.4
and a simula ion s udy is p esen ed in Sec ion 3.5. Finally, we include an appli-
ca ion o eal da a in Sec ion 3.6. The p oo s o he main esul s a e de e ed o
Sec ion 3.7.
3.2 Tes ing p ocedu e
The es ing p ocedu e is based on he compa ison o wo non-pa ame ic es ima o s
o he dis ibu ion o he esiduals Fεjin each popula ion. This in ol es non-
pa ame ic es ima ion o he loca ion and scale unc ions. All hese es ima o s
will be cons uc ed using he es ima o o he condi ional dis ibu ion unc ion
Fj(·|x) when he esponse is censo ed in oduced by Be an (1981):
ˆ
Fj(y|x) = 1 −Y
Zij ≤y,∆ij =1 Ã1−W(j)
ij (x, hn)
Pnj
l=1 I(Zlj ≥Zij)W(j)
lj (x, hn)!,
whe e
W(j)
ij (x, hn) = K((x−Xij)/hn)
Pnj
l=1 K((x−Xlj)/hn)
a e Nada aya-Wa son ype weigh s, Kis a known ke nel and hnis an app op ia e
bandwid h sequence.
Compa ison o eg ession cu es wi h censo ed esponses 61
Now conside he ollowing es ima o o he loca ion unc ion o each sample,
o j= 1, . . . , k,
ˆmj(x) = Z1
0
ˆ
F−1
j(s|x)J(s)ds,
and an es ima o o he common loca ion unc ion unde he null hypo hesis (which
we will deno e by m) aking in o accoun all he samples
ˆm(x) =
k
X
j=1
nj
n
ˆ
Xj(x)
ˆ
mix(x)ˆmj(x),
whe e,
ˆ
Xj(x) = 1
njhn
nj
X
i=1
Kµx−Xij
hn¶
is he ke nel es ima o o he densi y Xjo Xj, and
ˆ
mix(x) =
k
X
j0=1
nj0
nˆ
Xj0(x).
No e ha ˆ
Xj(x) can be compu ed in he usual way because he co a ia es do no
su e om censo ing. The es ima o o he scale unc ion σj om each sample is
ˆσ2
j(x) = Z1
0
ˆ
F−1
j(s|x)2J(s)ds −ˆm2
j(x).
The sco e unc ion Jwill be chosen so ha ˆmj(x) and ˆσ2
j(x) a e consis en ,
e en in he case o he ails o he Be an es ima o no being consis en .
Compu e he es ima o s o he censo ed esiduals in each sample
ˆ
Eij =Zij −ˆmj(Xij)
ˆσj(Xij)
o i= 1, . . . , nj,j= 1, . . . , k, and es ima e he dis ibu ion o he esiduals om
he censo ed sample ( ˆ
Eij,∆ij) using he Kaplan-Meie es ima o
ˆ
Fεj(y) = 1 −Y
ˆ
Eij ≤y,∆ij =1 Ã1−1
Pnj
l=1 I(ˆ
Elj ≥ˆ
Eij)!.(3.4)
62 Chap e 3
I he null hypo hesis is ue, we can es ima e he esiduals in each sample
using he es ima o o he common eg ession unc ion ˆm, ha is
ˆ
Eij0=Zij −ˆm(Xij)
ˆσj(Xij)
o i= 1, . . . , nj,j= 1, . . . , k, and es ima e he co esponding dis ibu ion om
he censo ed sample ( ˆ
Eij0,∆ij)
ˆ
Fεj0(y) = 1 −Y
ˆ
Eij0≤y,∆ij =1 Ã1−1
Pnj
l=1 I(ˆ
Elj0≥ˆ
Eij0)!.(3.5)
Unde he null hypo hesis, bo h ˆ
Fεjand ˆ
Fεj0a e es ima o s o Fεj. The ac
ha he e exis s some di e ence be ween hese wo es ima o s o he dis ibu ion
o he e o s gi es e idence o he inequali y o he loca ion unc ions. This idea
is o malized heo e ically in he ollowing heo em. No e ha ˆm(x) es ima es
consis en ly m(x) = Pk
j=1 pj
Xj(x)
mix(x)mj(x), whe e mix(x) = Pk
j=1 pj Xj(x) is he
mix u e o he densi ies o he co a ia es, p o ided ha nj/n →pj>0. Le
Fεj(y) = PµYj−mj(Xj)
σj(Xj)≤y¶
and
Fεj0(y) = PµYj−m(Xj)
σj(Xj)≤y¶
be he heo e ical e sions (wi hou es ima ed cu es) o he dis ibu ions conside-
ed in (3.4) and (3.5).
Theo em 3.1 Assume ha mjis con inuous, j= 1, . . . , k, and he momen s o
o de νo he dis ibu ions Fεj(y)and Fεj0(y)exis o all ν∈N. Then Fεj(y) =
Fεj0(y),−∞ < y < ∞,j= 1, . . . , k, i and only i m1(x) = . . . =mk(x) o all
x∈RX.
The equi alence gi en in he p e ious esul is a heo e ical jus i ica ion o he
p oposed es ing p ocedu e. I s p oo can be ound in Sec ion 3.7.
Compa ison o eg ession cu es wi h censo ed esponses 69
po hesis gi en in Co olla y 3.1 a e e y complica ed. He e we conside a boo s ap
p ocedu e based on he censo ed esiduals o app oxima e he c i ical alues.
Fi s , o j= 1, . . . , k and i= 1, . . . , nj, es ima e he censo ed esiduals in a
non pa ame ic way, using each sample sepa a ely
ˆ
Eij =Zij −ˆmj(Xij)
ˆσj(Xij).
F om he censo ed sample o es ima ed esiduals {(ˆ
Eij,∆ij), i = 1, . . . , nj}com-
pu e he Kaplan-Meie es ima o ˆ
Fεjand s anda dize hese esiduals in o de o
e i y he ini ial assump ion o ha ing loca ion unc ion 0 and scale unc ion 1.
The s anda dized esiduals a e ˜
Eij = ( ˆ
Eij −λ1j)/λ2j, whe e λ1j=Rˆ
F−1
εj(s)J(s)ds
and λ2j= (Rˆ
F−1
εj(s)2J(s)ds −λ2
1j)1/2.
Fo esampling he censo ed esiduals we use he ‘nai e boo s ap’ desc ibed
in E on (1981) and s udied in Ak i as (1986). Di e en app oaches o smoo h
boo s ap o censo ed da a we e conside ed in Gonz´alez-Man eiga, Cao and Ma -
on (1996).
The boo s ap p ocedu e we p opose consis s o he ollowing s eps. Fo ixed
Band o b= 1, . . . , B,
1. Fo each j= 1, . . . , k and i= 1, . . . , nj:
•Le Y∗
ij,b = ˆm(Xij)+ ˆσj(Xij)ε∗
ij,b, whe e ε∗
ij,b =Vij,b+ajSij,b,Vij,b is d awn
om ˆ
Fεj(s anda dized), and Sij,b is a andom a iable wi h mean ze o
and a iance one.
•Selec a andom a C∗
ij,b om a smoo hed e sion o ˆ
Gj(·|Xij), which is
he Be an es ima o o Gj(·|Xij) ob ained by eplacing ∆ij by 1 −∆ij
in he exp ession o ˆ
Fj(·|Xij).
•Le Z∗
ij,b = min{Y∗
ij,b, C∗
ij,b}and ∆∗
ij,b =I(Y∗
ij,b ≤C∗
ij,b).
2. The boo s ap samples a e, o j= 1, . . . , k,{(Xij, Z∗
ij,b,∆∗
ij,b), i = 1, . . . , nj}.
3. Le T∗
KS,b and T∗
CM,b be he es s a is ics ob ained om he boo s ap sam-
ples.

70 Chap e 3
I we deno e T∗
KS,(b) he b- h o de s a is ic o he alues T∗
KS,1, . . . , T∗
KS,B ob-
ained in s ep 3 (analogously o T∗
CM,(b)), hen T∗
KS,([(1−α)B]) and T∗
CM,([(1−α)B])
app oxima e he (1 −α)-quan iles o he dis ibu ion o TKS and TCM unde he
null hypo hesis espec i ely.
3.5 Simula ion s udy
In his sec ion we p esen some simula ions in o de o s udy he p ac ical be-
ha io o he p oposed boo s ap p ocedu e. We es ic ou sel es o wo-sample
si ua ions (k= 2). Mo e p ecisely, we conside he ollowing models:
(i)m1(x) = x;m2(x) = x
(ii)m1(x) = exp(x); m2(x) = exp(x)
(iii)m1(x) = x;m2(x) = x+ 1
(i )m1(x) = exp(x); m2(x) = exp(x) + x
Clea ly, models (i) and (ii) co espond o he null hypo hesis and models (iii) and
(i ) o he al e na i e hypo hesis. In each case we conside a homoscedas ic and
a he e oscedas ic si ua ion. In he homoscedas ic case he a iances a e
σ2
1(x) = 0.25 and σ2
2(x) = 0.50, (3.8)
while in he he e oscedas ic case he a iance unc ions a e
σ2
1(x) = ex
R1
0e d and σ2
2(x) = e2x
R1
0e2 d .(3.9)
No e ha in he he e oscedas ic case he a iances a e la ge han in he ho-
moscedas ic case.
The censo ing a iables a e Cj=mj(Xj) + σj(Xj)ρj, whe e ρjhas su i al
unc ion 1 −Fρ(y) = (1 −Fε(y))β. This mechanism o censo ing can be seen as
a ‘condi ional Koziol-G een model’ (see Koziol and G een, 1976) and i allows
us o ha e he same amoun o censo ing o e all he suppo o he co a ia es.
The expec ed p opo ion o censo ed da a is (1 + β)−1. In he ables we conside
β= 1/3 (25% o censo ing) and β= 1 (50% o censo ing).
Compa ison o eg ession cu es wi h censo ed esponses 71
In he heo e ical esul s we ha e used only one bandwid h. As in Sec ion 2.5,
we ha e ound ha he bandwid h has no a big impac on he esul s o he
es s, bu i is ecommendable o use he same bandwid h o es ima e mand mj.
The a iance unc ions could be es ima ed wi h di e en bandwid hs. In hese
simula ions we use a bandwid h o he o m h=cn−3/10 o es ima e m,mjand σj,
o j= 1,2. The bandwid hs chosen in his way e i y he egula i y condi ions
assumed in he heo e ical esul s. In he ables he cases c= 1 and c= 1.5 a e
shown. This will allow us o check he sensi i i y o he es o he change o he
bandwid h. Fo he ke nel needed o calcula e he weigh s ha appea in he Be an
es ima o , we choose he Epanechniko ke nel K(u) = 0.75(1 −u2)I(|u|<1).
In Tables 3.1 and 3.2 he dis ibu ion o he e o s is Exponen ial, ans o med
such ha R1
0F−1
εj(s)J(s)ds = 0 and R1
0F−1
εj(s)2J(s)ds = 1 and he co a ia es a e
uni o mly dis ibu ed in [0,1]. The eg ession and a iance unc ions a e hose
co esponding o exp essions (3.1) and (3.2) wi h he choice J(s)=0.75−1I(0 ≤
s≤0.75) o he sco e unc ion. Fo he es s a is ics in (3.6) and (3.7) we ake as
he h eshold T he alue co esponding o he quan ile 75% o he combined sam-
ple o he es ima ed esiduals unde he null hypo hesis. No e ha all hese choices
a e easonable o he models and censo ing mechanisms we ha e conside ed. We
wo k wi h aj=n−3/10
jin he smoo h boo s ap.
Table 3.1 shows he p opo ion o ejec ions in 1000 ials o sample sizes
(n1, n2) = (50,50), (100,50) and (100,100), and when he expec ed amoun o
censo ed da a is 25%. Table 3.2 shows he p opo ion o ejec ions in 1000 ials
o sample sizes (n1, n2) = (100,100), (200,100) and (200,200) when he expec ed
amoun o censo ed da a is 50%. In all cases we wo ked wi h B= 200 boo s ap
eplica ions and signi icance le els α= 0.05 and α= 0.10. La ge samples sizes o
models wi h 50% o censo ed da a a e jus i ied by he di icul y o hose models.
The app oxima ion o he le el –models (i) and (ii)– is good in mos cases.
The esul s o models (iii) and (i ) show ha he es s gain powe as he sample
sizes inc ease. In almos all cases he es based on TCM gi es be e esul s han
he es based on TKS, and we also obse e ha he choice o he bandwid h has
li le impac on he ejec ion p obabili ies.
72 Chap e 3
Table 3.1: Rejec ion p obabili ies unde models (i) o (i ) o he es s based on
TKS and TCM when he expec ed amoun o censo ed da a is 25%. The models a e
homoscedas ic, wi h a iances gi en in (3.8), and he e oscedas ic, wi h a iances
gi en in (3.9).
c= 1 c= 1.5
TKS TCM TKS TCM
(n1, n2)α: 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
Homoscedas ic models
(50,50) (i) 0.049 0.084 0.048 0.093 0.044 0.095 0.051 0.091
(ii) 0.042 0.089 0.048 0.094 0.044 0.090 0.051 0.094
(iii) 0.984 0.993 0.989 0.996 0.984 0.994 0.989 0.994
(i ) 0.505 0.629 0.550 0.674 0.489 0.643 0.560 0.667
(100,50) (i) 0.059 0.112 0.057 0.104 0.061 0.098 0.067 0.098
(ii) 0.067 0.107 0.058 0.102 0.055 0.096 0.058 0.100
(iii) 0.999 0.999 0.998 0.999 1.000 1.000 1.000 1.000
(i ) 0.557 0.724 0.541 0.690 0.580 0.720 0.568 0.699
(100,100) (i) 0.055 0.109 0.059 0.103 0.056 0.109 0.058 0.102
(ii) 0.060 0.117 0.062 0.103 0.057 0.105 0.058 0.110
(iii) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(i ) 0.863 0.924 0.888 0.935 0.862 0.923 0.879 0.932
He e oscedas ic models
(50,50) (i) 0.045 0.090 0.044 0.094 0.044 0.085 0.046 0.090
(ii) 0.042 0.085 0.043 0.089 0.045 0.090 0.048 0.089
(iii) 0.771 0.849 0.792 0.864 0.763 0.853 0.777 0.866
(i ) 0.243 0.360 0.256 0.357 0.245 0.346 0.257 0.356
(100,50) (i) 0.048 0.087 0.044 0.086 0.051 0.102 0.048 0.092
(ii) 0.050 0.086 0.046 0.088 0.048 0.089 0.049 0.091
(iii) 0.909 0.955 0.880 0.939 0.913 0.956 0.892 0.947
(i ) 0.236 0.343 0.191 0.283 0.246 0.363 0.221 0.313
(100,100) (i) 0.066 0.112 0.057 0.116 0.059 0.117 0.059 0.111
(ii) 0.061 0.111 0.059 0.112 0.061 0.109 0.056 0.110
(iii) 0.976 0.987 0.973 0.989 0.971 0.985 0.974 0.988
(i ) 0.472 0.582 0.476 0.583 0.465 0.584 0.474 0.581
Compa ison o eg ession cu es wi h censo ed esponses 73
Table 3.2: Rejec ion p obabili ies unde models (i) o (i ) o he es s based on
TKS and TCM when he expec ed amoun o censo ed da a is 50%. The models a e
homoscedas ic, wi h a iances gi en in (3.8), and he e oscedas ic, wi h a iances
gi en in (3.9).
c= 1 c= 1.5
TKS TCM TKS TCM
(n1, n2)α: 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
Homoscedas ic models
(100,100) (i) 0.052 0.093 0.055 0.093 0.042 0.081 0.049 0.083
(ii) 0.051 0.083 0.050 0.099 0.037 0.075 0.046 0.087
(iii) 0.995 0.997 0.995 0.997 0.995 0.997 0.994 0.996
(i ) 0.751 0.855 0.813 0.903 0.723 0.838 0.792 0.888
(200,100) (i) 0.072 0.126 0.078 0.137 0.055 0.108 0.057 0.116
(ii) 0.071 0.119 0.074 0.139 0.064 0.111 0.064 0.114
(iii) 1.000 1.000 0.999 1.000 0.999 1.000 1.000 1.000
(i ) 0.808 0.903 0.841 0.910 0.851 0.922 0.866 0.933
(200,200) (i) 0.053 0.101 0.059 0.105 0.052 0.090 0.057 0.098
(ii) 0.055 0.094 0.059 0.103 0.055 0.092 0.058 0.098
(iii) 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(i ) 0.980 0.992 0.988 0.993 0.982 0.991 0.986 0.993
He e oscedas ic models
(100,100) (i) 0.032 0.067 0.042 0.089 0.038 0.077 0.043 0.089
(ii) 0.035 0.070 0.041 0.081 0.039 0.071 0.045 0.086
(iii) 0.932 0.964 0.942 0.972 0.905 0.953 0.926 0.966
(i ) 0.387 0.520 0.441 0.578 0.370 0.495 0.430 0.565
(200,100) (i) 0.050 0.103 0.065 0.125 0.045 0.098 0.055 0.104
(ii) 0.053 0.091 0.062 0.122 0.046 0.092 0.052 0.110
(iii) 0.987 0.995 0.991 0.997 0.990 0.997 0.990 0.998
(i ) 0.381 0.500 0.366 0.478 0.381 0.513 0.381 0.512
(200,200) (i) 0.044 0.081 0.049 0.090 0.041 0.082 0.048 0.082
(ii) 0.046 0.086 0.049 0.089 0.046 0.086 0.046 0.089
(iii) 0.998 0.999 0.998 1.000 0.998 1.000 0.998 1.000
(i ) 0.688 0.787 0.722 0.812 0.669 0.786 0.722 0.800
74 Chap e 3
3.6 Applica ion o eal da a
We illus a e ou es ing p ocedu e wi h an applica ion o he Small Cell Lung
Cance Da a. The da a se is a ailable in Ying, Jung and Wei (1995) and consis s
o li e imes o pa ien s su e ing om small cell lung cance . The pa ien s we e
di ided in o wo g oups which ollowed wo di e en ea men s (G oup A and
G oup B). The i s g oup consis ed o 62 pa ien s (15 censo ed) and he second
g oup consis ed o 59 pa ien s (8 censo ed). We conside ed he base 10 log o he
su i al ime (in days) as esponse a iable and he age as co a ia e. The suppo
o he co a ia e was ans o med in o he in e al [0,1]. We wo ked wi h di e en
alues o he bandwid h needed in he es ima ion, anging om 0.15 o 0.40.
We ha e pe o med he es o equali y o he eg ession cu es o he wo cu es
using as sco e unc ion J(s)=0.75−1I(0 ≤s≤0.75) and J(s) = 0.50−1I(0.25 ≤
s≤0.75). The second choice o he unc ion Jp oduces cu es close o he con-
di ional median. The ob ained esul s a e e y simila . The p- alues a e ob ained
om 1000 boo s ap eplica ions. Figu e 3.1 shows he es ima ed cu es, using
h= 0.30 as a bandwid h.
When es ing o he equali y o he cu es, he null hypo hesis is clea ly e-
jec ed in all cases, wi h p- alues smalle han 0.02 o he s a is ic TKM and smalle
han 0.005 o TCM . Howe e , i seems easonable o suppose ha he eg ession
cu es di e only by a shi (see Figu e 3.1). A es o check ha can be pe -
o med by ans o ming he esponse a iables in Z0
ij =Zij − j, o j= 1,2 and
i= 1, . . . , nj, whe e j=n−1P2
l=1 Pnl
i=1 ˆmj(Xil). In his case he p- alues a e
la ge han 0.55 o he s a is ic TKM and la ge han 0.67 o TCM . All hese
esul s a e summa ized in Figu es 3.2 and 3.3, which show g aphs o he p- alues
e sus he bandwid h.

Compa ison o eg ession cu es wi h censo ed esponses 75
age
log (su i al ime)
0.0 0.2 0.4 0.6 0.8 1.0
1.5 2.0 2.5 3.0 3.5
age
log (su i al ime)
0.0 0.2 0.4 0.6 0.8 1.0
1.5 2.0 2.5 3.0 3.5
Figu e 3.1: Sca e plo o ‘log10(su i al ime)’ e sus ‘age’ ( escaled o [0,1]) and
es ima ed eg ession cu es o G oup A (solid line, + o uncenso ed da a, ¤ o
censo ed da a) and G oup B (dashed line, × o uncenso ed da a, 4 o censo ed
da a), wi h J(s)=0.75−1I(0 ≤s≤0.75) ( op) and J(s) = 0.50−1I(0.25 ≤s≤
0.75) (bo om).
76 Chap e 3
h
p- alue
0.15 0.20 0.25 0.30 0.35 0.40
0.0 0.01 0.02 0.03 0.04 0.05 0.06
h
p- alue
0.15 0.20 0.25 0.30 0.35 0.40
0.0 0.01 0.02 0.03 0.04 0.05 0.06
Figu e 3.2: G aphs o he p- alues as unc ion o he bandwid h hwhen es ing o
he equali y o he eg ession cu es wi h he es s a is ics TKS (line wi h ci cles)
and TCM (line wi h c osses). The loca ion cu es co espond o J(s) = 0.75−1I(0 ≤
s≤0.75) (le ) and J(s) = 0.50−1I(0.25 ≤s≤0.75) ( igh ). The solid ho izon al
line co esponds o a p- alue o 0.05.
h
p- alue
0.15 0.20 0.25 0.30 0.35 0.40
0.0 0.2 0.4 0.6 0.8 1.0
h
p- alue
0.15 0.20 0.25 0.30 0.35 0.40
0.0 0.2 0.4 0.6 0.8 1.0
Figu e 3.3: G aphs o he p- alues as unc ion o he bandwid h hwhen es ing
o cons an di e ence be ween he eg ession cu es wi h he es s a is ics TKS
(line wi h ci cles) and TCM (line wi h c osses). The loca ion cu es co espond o
J(s)=0.75−1I(0 ≤s≤0.75) (le ) and J(s)=0.50−1I(0.25 ≤s≤0.75) ( igh ).
The solid ho izon al line co esponds o a p- alue o 0.05.
Compa ison o eg ession cu es wi h censo ed esponses 77
3.7 Auxilia y esul s and p oo s
P oo o Theo em 3.1. Assume ha Fεj(y) = Fεj0(y), o −∞ < y < ∞. We
w i e
PµYj−m(Xj)
σj(Xj)≤y¶=PµYj−mj(Xj)
σj(Xj)+mj(Xj)−m(Xj)
σj(Xj)≤y¶,
o all y, o equi alen ly
Pµexp ½Yj−m(Xj)
σj(Xj)¾≤y¶
=Pµexp ½Yj−mj(Xj)
σj(Xj)¾exp ½mj(Xj)−m(Xj)
σj(Xj)¾≤y¶,
o all y. Since (Yj−mj(Xj))/σj(Xj) and Xja e independen , i ollows ha
E"µexp ½Yj−m(Xj)
σj(Xj)¾¶2ν#
=E"µexp ½Yj−mj(Xj)
σj(Xj)¾¶2ν#E"µexp ½mj(Xj)−m(Xj)
σj(Xj)¾¶2ν#,
o all ν. Then
E"µexp ½mj(Xj)−m(Xj)
σj(Xj)¾¶2ν#= 1,
o all ν. Ca leman’s condi ion (see e.g. Felle , 1966) ensu es ha
Pµexp ½mj(Xj)−m(Xj)
σj(Xj)¾= 1¶= 1
o
Pµmj(Xj)−m(Xj)
σj(Xj)= 0¶= 1,
and his clea ly implies ha mj(x) = m(x) o all j= 1, . . . , k and o all x∈RX,
excep o a se o poin s o p obabili y ze o. The con inui y allows us o ex end
he equali y o he eg ession cu es o he whole suppo o he co a ia es. The
con e se implica ion is i ial.
Fi s we se ou auxilia y lemmas, and hen we p o e he main esul s.
78 Chap e 3
Lemma 3.1 Assume (A1)-(A5) and Hej(y|x)sa is y (A6). Then unde he null
hypo hesis H0, o j= 1, . . . , k,
ˆ
Hej0(y)−Hej(y)
=1
nj
nj
X
i=1
I(Eij ≤y)−Hej(y)−1
nj
nj
X
i=1
yhej(y|Xij)ζj(Zij,∆ij|Xij)
−1
n
k
X
l=1
nl
X
i=1
hej(y|Xil) Xj(Xil)
mix(Xil)
σl(Xil)
σj(Xil)ηl(Zil,∆il|Xil) + oP(n−1/2),
uni o mly in −∞ < y ≤T.
P oo . F om he p oo o P oposi ion A.2 in Van Keilegom and Ak i as (1999),
we ha e ha
ˆ
Hej0(y)−Hej(y) = 1
nj
nj
X
i=1
I(Eij ≤y)−Hej(y) (3.10)
+Zhej(y|x)ˆm(x)−m(x)
σj(x) Xj(x)dx +Zyhej(y|x)ˆσj(x)−σj(x)
σj(x) Xj(x)dx
+oP(n−1/2
j),
uni o mly in −∞ < y ≤T. The las e m is oP(n−1/2
j) because o he uni o m
consis ency o ˆmand ˆσj. The consis ency o ˆσjis gi en in P oposi ion 4.5 in
Van Keilegom and Ak i as (1999). The consis ency o ˆmcan be ob ained using
he consis ency o ˆml(also gi en in P oposi ion 4.5 in Van Keilegom and Ak i as,
1999), he consis ency o ˆ
Xland ˆ
mix and aking in o accoun he ela ion
ˆm(x)−m(x) =
k
X
l=1
nl
n
ˆ
Xl(x)
ˆ
mix(x)( ˆml(x)−m(x))
=
k
X
l=1
nl
n
Xl(x)
mix(x)( ˆml(x)−m(x)) + oP(n−1/2),
uni o mly in x.
Fi s using P oposi ion 4.8 in Van Keilegom and Ak i as (1999)
ˆm(x)−m(x)
=−1
nhn
1
mix(x)
k
X
l=1
nl
X
i=1
σl(x)Kµx−Xil
hn¶ηl(Zil,∆il|x) + oP(n−1/2),
Compa ison o eg ession cu es wi h censo ed esponses 85
conside a ions. Unde Hl.a., he es ima o o he common eg ession cu e ˆm
es ima es mn(x) = m0(x) + n−1/2R(x), whe e R(x) = Pk
l=1 pl
Xl(x)
mix(x) l(x), and
ˆmj(x) es ima es mjn(x) = m0(x)+n−1/2 j(x). The censo ed esiduals wi h espec
o mna e
Ej0=Zj−mn(Xj)
σj(Xj),
and wi h espec o mjn he esiduals a e
Ej=Zj−mjn(Xj)
σj(Xj).
We ha e he ela ion
Ej0=Zj−mn(Xj)
σj(Xj)=Ej+mjn(Xj)−mn(Xj)
σj(Xj)=Ej−n−1/2R(Xj)− j(Xj)
σj(Xj).
No e also ha now ˆ
Fεj0(y) es ima es Fεj0(y) = P((Yj−mn(Xj))/σj(Xj)≤y).
We deno e Hej0(y) = P(Ej0≤y), Hej01(y) = P(Ej0≤y, ∆j= 1), Hej0(y|x) =
P(Ej0≤y|Xj=x), Hej01(y|x) = P(Ej0≤y, ∆j= 1|Xj=x).
I we de ine E0
j= (Z0
j−m0(Xj))/σj(Xj) and deno e H0
ej(y|x) = P(E0
j≤
y|Xj=x) and H0
ej1(y|x) = P(E0
j≤y, ∆j= 1|Xj=x), hen i is easy o check
ha Hej(y|x) = H0
ej(y|x) and Hej1(y|x) = H0
ej1(y|x), which do no depend on n.
No e ha he independence o Yjand Cjgi en Xj=ximplies he independence
o Y0
jand C0
jgi en Xj=x.
Lemma 3.5 Assume (A1)-(A5) and Hej(y|x)sa is y (A6) and (AR) holds. Then,
unde he al e na i e hypo hesis Hl.a., o any j= 1, . . . , k,
ˆ
Hej0(y)−Hej0(y)
=1
nj
nj
X
i=1
I(Eij ≤y)−Hej(y)−1
nj
nj
X
i=1
yhej(y|Xij)ζj(Zij,∆ij|Xij)
−1
n
k
X
l=1
nl
X
i=1
hej(y|Xil) Xj(Xil)
mix(Xil)
σl(Xil)
σj(Xil)ηl(Zil,∆il|Xil) + oP(n−1/2
j),
uni o mly in −∞ < y ≤T.

86 Chap e 3
P oo . We ollow he p oo o P oposi ion A.2 in Van Keilegom and Ak i as
(1999) and w i e
ˆ
Hej0(y)−Hej0(y) = n−1
j
nj
X
i=1
I(Eij0≤y)−Hej0(y) (3.13)
+ZHej0µyˆσj(x) + ˆm(x)−mn(x)
σj(x)¯¯¯¯
x¶ Xj(x)dx
−ZHej0(y|x) Xj(x)dx +oP(n−1/2),
uni o mly in −∞ < y ≤T. No e ha he emainde e m in (3.13) is oP(n−1/2)
p o ided ha ˆm−mnsa is ies P oposi ions 4.5, 4.6 and 4.7 in Van Keilegom and
Ak i as (1999). These p oposi ions can be shown o hold ue unde s anda d lines
o p oo .
We will analyze in de ail each e m o he exp ession abo e. Fi s
Hej0(y) = ZHej0(y|x) Xj(x)dx (3.14)
=ZHej(y|x) Xj(x)dx +Zhej(y|x)x)mn(x)−mjn(x)
σj(x) Xj(x)dx+
=Hej(y) + n−1/2E·hej(y|Xj)R(Xj)− j(Xj)
σj(Xj)¸+o(n−1/2),
and
ZHej0µyˆσj(x) + ˆm(x)−mn(x)
σj(x)¯¯¯¯
x¶ Xj(x)dx (3.15)
=ZHej(y|x) Xj(x)dx
+Zhej(y|x)yˆσj(x) + ˆm(x)−mjn(x)−yσj(x)
σj(x) Xj(x)dx +o(n−1/2)
=Hej(y) + Zhej(y|x)y(ˆσj(x)−σj(x)) + ˆm(x)−mn(x)
σj(x) Xj(x)dx
+n−1/2E·hej(y|Xj)R(Xj)− j(Xj)
σj(Xj)¸+o(n−1/2).
An applica ion o he p oo o Lemma A.1 in Van Keilegom and Ak i as (1999)
Compa ison o eg ession cu es wi h censo ed esponses 87
yields
sup
y¯¯¯¯¯
n−1
j
nj
X
i=1 ½IµEij −n−1/2R(Xij)− j(Xij)
σj(Xij)≤y¶
−PµEj−n−1/2R(Xj)− j(Xj)
σj(Xj)≤y¶−I(Eij ≤y) + P(Ej≤y)¾¯¯¯¯
=oP(n−1/2
j).
Conside ing he ollowing p obabili y as a unc ion o yand using a Taylo
expansion we ob ain
PµEj−n−1/2R(Xj)− j(Xj)
σj(Xj)≤y¶
=ZPµEj−n−1/2R(Xj)− j(Xj)
σj(Xj)≤y¯¯¯¯
Xj=x¶ Xj(x)dx
=ZP(Ej≤y|Xj=x) Xj(x)dx +n−1/2E·hej(y|Xj)R(Xj)− j(Xj)
σj(Xj)¸
+o(n−1/2)
=P(Ej≤y) + n−1/2E·hej(y|Xj)R(Xj)− j(Xj)
σj(Xj)¸+o(n−1/2),
and hence
n−1
j
nj
X
i=1
I(Eij0≤y) = n−1
j
nj
X
i=1
IµEij −n−1/2R(Xij)− j(Xij)
σj(Xij)≤y¶(3.16)
=n−1
j
nj
X
i=1
I(Eij ≤y) + n−1/2E·hej(y|Xj)R(Xj)− j(Xj)
σj(Xj)¸+oP(n−1/2).
Subs i u ing (3.14), (3.15) and (3.16) in (3.13), we ob ain
ˆ
Hej0(y)−Hej0(y) = n−1
j
nj
X
i=1
I(Eij ≤y)−Hej(y)
+Zhej(y|x)y(ˆσj(x)−σj(x)) + ˆm(x)−mn(x)
σj(x) Xj(x)dx +oP(n−1/2).
Since ˆm(x)−mn(x) = Pk
l=1
nl
n
Xl(x)
mix(x)( ˆml(x)−mln(x)) + oP(n−1/2), he in eg al on
he igh -hand side o he exp ession abo e can be handled in a simila way as in
he p oo o Lemma 3.1.
88 Chap e 3
Lemma 3.6 Assume (A1)-(A5) and Hej1(y|x)sa is y (A6) and (AR) holds. Then,
unde he al e na i e hypo hesis Hl.a., o any j= 1, . . . , k,
ˆ
Hej10(y)−Hej10(y)
=1
nj
nj
X
i=1
I(Eij ≤y, ∆ij = 1) −Hej1(y)−1
nj
nj
X
i=1
yhej1(y|Xij)ζj(Zij,∆ij|Xij)
−1
n
k
X
l=1
nl
X
i=1
hej1(y|Xil) Xj(Xil)
mix(Xil)
σl(Xil)
σj(Xil)ηl(Zil,∆il|Xil) + oP(n−1/2
j),
uni o mly in −∞ < y ≤T.
P oo . Simila o he p oo o Lemma 3.5.
P oo o Theo em 3.4. Fi s we w i e
Zy
−∞
d(ˆ
Hej10(s)−Hej10(s))
1−Hej0(s)=Zy
−∞
d(ˆ
Hej10(s)−Hej10(s))
1−Hej(s)(3.17)
+Zy
−∞ µ1
1−Hej0(s)−1
1−Hej(s)¶d(ˆ
Hej10(s)−Hej10(s)).
The p oo o Co olla y A.5 in Van Keilegom and Ak i as (1999) can be adap ed
he e o show ha
sup
−∞<y≤T¯¯¯¯Zy
−∞ µ1
1−Hej0(s)−1
1−Hej(s)¶d(ˆ
Hej10(s)−Hej10(s))¯¯¯¯
=oP(n−1/2).
(3.18)
Indeed, equa ion (3.14) in he p oo o Lemma 3.5 es ablishes ha Hej0(y)−
Hej(y) = O(n−1/2). No e ha his o de is no s ochas ic and be e han he
equi alen one needed in he abo e-men ioned p oo . I su ices o ollow he same
s eps o ob ain (3.18). Hence he las e m o he exp ession (3.17) is oP(n−1/2),
and we ob ain
Zy
−∞
d(ˆ
Hej10(s)−Hej10(s))
1−Hej0(s)=Zy
−∞
d(ˆ
Hej10(s)−Hej10(s))
1−Hej(s)+oP(n−1/2).(3.19)
Simila ly o equa ion (3.14), i holds ha
Hej10(y) = ZHej1µyσj(x) + mn(x)−mjn(x)
σj(x)¯¯¯¯
x¶ Xj(x)dx,
Compa ison o eg ession cu es wi h censo ed esponses 89
and aking de i a i es and a Taylo expansion o hej1a ound y
hej10(y) = Zhej1µyσj(x) + mn(x)−mjn(x)
σj(x)¯¯¯¯
x¶ Xj(x)dx
=hej1(y) + O(n−1/2).
I ollows ha
hej10(s)
(1 −Hej0(s))2=hej1(s)
(1 −Hej(s))2+hej10(s)µ1
(1 −Hej0(s))2−1
(1 −Hej(s))2¶
+1
(1 −Hej(s))2(hej10(s)−hej1(s))
=hej1(s)
(1 −Hej(s))2+O(n−1/2),
and since ˆ
Hej0(y)−Hej0(y) = OP(n−1/2), we ob ain
Zy
−∞
ˆ
Hej0(s)−Hej0(s)
(1 −Hej0(s))2dHej10(s) = Zy
−∞
ˆ
Hej0(s)−Hej0(s)
(1 −Hej(s))2dHej1(s) + oP(n−1/2).
(3.20)
Using (3.19) and (3.20), as in he p oo o Theo em 3.2, we ha e ha
ˆ
Fεj0(y)−Fεj0(y)
= (1 −Fεj0(y)) "Zy
−∞
ˆ
Hej0(s)−Hej0(s)
(1 −Hej(s))2dHej1(s) + Zy
−∞
d(ˆ
Hej10(s)−Hej10(s))
1−Hej(s)#
+oP(n−1/2).
F om he p oo o Theo em 3.2, we also ha e
ˆ
Fεj(y)−Fεj(y)
= (1 −Fεj(y)) "Zy
−∞
ˆ
Hej(s)−Hej(s)
(1 −Hej(s))2dHej1(s) + Zy
−∞
d(ˆ
Hej1(s)−Hej1(s))
1−Hej(s)#
+oP(n−1/2
j).
90 Chap e 3
Now w i e
Fεj0(y) = PµYj−mn(Xj)
σj(Xj)≤y¶
=PµYj−mjn(Xj)
σj(Xj)−n−1/2R(Xj)− j(Xj)
σj(Xj)≤y¶
=ZPµYj−mjn(Xj)
σj(Xj)−n−1/2R(Xj)− j(Xj)
σj(Xj)≤y¯¯¯¯
Xj=x¶ Xj(x)dx.
I we conside he p obabili y inside he in eg al as a unc ion o yand apply a
Taylo expansion, we ob ain
Fεj0(y) = Fεj(y) + n−1/2 εj(y)E·R(Xj)− j(Xj)
σj(Xj)¸+o(n−1/2).(3.21)
S aigh o wa d calcula ions lead o ηj(Zj,∆j|Xj) = η0
j(Z0
j,∆j|Xj). Following
he same s eps as in he p oo o Theo em 3.2, using Lemmas 3.5 and 3.6, and
aking in o accoun ha Fεj0(y) = Fεj(y) + O(n−1/2), we ob ain he exp essions
ˆ
Fεj0(y)−Fεj0(y) (3.22)
=1
nj
nj
X
i=1
ξej(Eij,∆ij, y)−1
nj
nj
X
i=1
(1 −Fεj(y))ζj(Zij,∆ij|Xij)γj2(y|Xij)
−1
n
k
X
l=1
nl
X
i=1
(1 −Fεj(y)) Xj(Xil)
mix(Xil)
σl(x)
σj(Xil)η0
l(Z0
il,∆il|Xil)γj1(y|Xil) + oP(n−1/2),
and
ˆ
Fεj(y)−Fεj(y) (3.23)
=1
nj
nj
X
i=1
ξej(Eij,∆ij, y)−1
nj
nj
X
i=1
(1 −Fεj(y))ζj(Zij,∆ij|Xij)γj2(y|Xij)
−1
nj
nj
X
i=1
(1 −Fεj(y))η0
j(Z0
ij,∆ij|Xij)γj1(y|Xij) + oP(n−1/2
j).
Finally, by combining exp essions (3.21), (3.22) and (3.23) we ob ain he ep esen-
a ion gi en in he s a emen o he Theo em. The leading e m o he ob ained
ep esen a ion does no depend on n, because he unc ions η0
ja e de ined in e ms
o dis ibu ions o andom a iables which do no depend on nand he unc ions

Compa ison o eg ession cu es wi h censo ed esponses 91
γj1a e de ined in e ms o dis ibu ions o esiduals Ej, which do no depend on
nei he .
P oo o Theo em 3.5. The leading e m o he ep esen a ion gi en in Theo em
3.4 when wo king unde Hl.a.
n1/2
k
X
l=1
pl(n−1
l
nl
X
i=1
ψ0
jl(Xil, Z0
il,∆il, y))
equals he leading e m o he ep esen a ion gi en unde H0in Theo em 3.2
n1/2
k
X
l=1
pl(n−1
l
nl
X
i=1
ψjl(Xil, Zil,∆il, y)),
whe e m0in he i s exp ession abo e plays he ole o min he second one. Hence
he asymp o ic beha io is he same and he weak con e gence ollows immedia ely.
P oo o Co olla y 3.2. The con e gence o he es s a is ics unde he al e -
na i e hypo hesis Hl.a. can be ob ained in he same way as he p oo o Co olla y
3.1, simply by aking in o accoun he weak con e gence o he p ocess es ablished
in Theo em 3.5.
Chap e 4
Goodness-o - i es s o
pa ame ic models in censo ed
eg ession
In his chap e we in oduce a goodness-o - i es o pa ame ic eg ession models
when he esponse a iable is igh censo ed. The es is based on he compa ison
o a pa ame ic es ima o and a nonpa ame ic es ima o o he dis ibu ion o he
e o s. Kolmogo o -Smi no and C am´e - on Mises ype s a is ics a e p oposed.
A boo s ap mechanism is used o app oxima e he c i ical alues o he es . Some
simula ions a e included and a eal da a se is analyzed.
This chap e is based on Pa do-Fe n´andez, Van Keilegom and Gonz´alez-Man-
eiga (2005b).
4.1 In oduc ion and s a is ical model
As explained in Chap e 1, in many p ac ical si ua ions pa ame ic eg ession
models a e appealing because hey desc ibe he ela ionship be ween he esponse
and he co a ia e in a simple way and allow o in e p e abili y o he pa ame e s.
Ne e heless, i he pa ame ic model ails hen he conclusions will be e oneous.
Any pa ame ic analysis should be accompanied by a es o check i s alidi y and
93
94 Chap e 4
a oid misspeci ica ion and w ong conclusions. This mo i a es he de elopmen o
speci ic goodness-o - i es s o pa ame ic models in eg ession.
In he con ex o censo ed da a he s a is ical model can be desc ibed as ollows.
Le (X, Y ) be a andom ec o , whe e Y ep esen s a ce ain esponse a iable
associa ed wi h he co a ia e X. Assume ha he esponse a iable Yis subjec
o andom igh censo ing. This means ha he e exis s a censo ing a iable C,
independen o Ygi en X, such ha we obse e Z= min{Y, C}and he indica o
o censo ing ∆ = I(Y≤C). Conside he ollowing non-pa ame ic eg ession
model:
Y=m(X) + σ(X)ε.
The e o a iable εis independen o X,mis an unknown condi ional loca ion
unc ion
m(x) = Z1
0
F−1(s|x)J(s)ds
and σis an unknown condi ional scale unc ion ep esen ing possible he e oscedas-
ici y
σ2(x) = Z1
0
F−1(s|x)2J(s)ds −m2(x),
whe e F(·|x) is he condi ional dis ibu ion o Ygi en he alue xo he co a ia e
X,F−1(s|x) = in {y;F(y|x)≥s}is he co esponding quan ile unc ion and
J(s) is a sco e unc ion sa is ying R1
0J(s)ds = 1 (in gene al, o any dis ibu ion
unc ion Fwe deno e F−1(s) = in {y;F(y)≥s} o he co esponding quan ile
unc ion and τF= in {y;F(y) = 1}). Le Fεbe he dis ibu ion o he e o ε. By
cons uc ion R1
0F−1
ε(s)J(s)ds = 0 and R1
0F−1
ε(s)2J(s)ds = 1. The sample consis s
o nindependen eplica ions (Xi, Zi,∆i), i= 1, . . . , n, om he dis ibu ion o
(X, Z, ∆).
We ecall he e ha he choice o he unc ion Jleads o di e en loca ion and
scale unc ions. An in e es ing choice is J(s)=(q−p)−1I(p≤s≤q), o some
0≤p < q ≤1, which leads o immed means and immed a iances.
Gi en a pa icula pa ame ic class o eg ession unc ions
M={mθ;θ∈Θ},
Goodness-o - i es s in censo ed eg ession 101
(ii) mθ(x) is wice con inuously di e en iable wi h espec o θ o all x∈RX.
Assuming (A5) and he ep esen a ion gi en in (4.4), a Taylo expansion o
mθ(u) as unc ion o θa ound θ0leads o
mˆ
θ(u)−mθ0(u) = n−1
n
X
i=1
ϕθ0(u, Xi, Zi,∆i) + oP(n−1/2),
whe e
ϕθ0(u, x, z, δ) = Ã∂mθ(u)
∂θ1¯¯¯¯θ=θ0
,..., ∂mθ(u)
∂θp¯¯¯¯θ=θ0!~ω(x, z, δ).(4.5)
Theo em 4.2 Assume (A1)-(A5). Then, unde he null hypo hesis H0,
ˆ
Fε0(y)−ˆ
Fε(y) = (1 −Fε(y))n−1
n
X
i=1
ψθ0(Xi, Zi,∆i, y) + oP(n−1/2),
uni o mly in −∞ < y ≤T, whe e
ψθ0(x, z, δ, y) = Zσ−1(u)ϕθ0(u, x, z, δ) X(u)γ0(y|u)du +γ0(y|x)η(z, δ|x).
Theo em 4.3 Assume (A1)-(A5). Then, unde he null hypo hesis H0, he p ocess
ˆ
W(y) = n1/2(ˆ
Fε0(y)−ˆ
Fε(y)),−∞ < y ≤T, con e ges weakly o a cen e ed
Gaussian p ocess W(y)wi h co a iance unc ion
Co (W(y), W(y0)) = (1 −Fε(y))(1 −Fε(y0))E(ψθ0(X, Z, ∆, y), ψθ0(X, Z, ∆, y0)).
Co olla y 4.1 Assume (A1)-(A5). Then, unde he null hypo hesis H0,
TKS
d
→sup
−∞<y<T |W(y)|,
TCM
d
→ZT
−∞
W2(y)dFε(y).

102 Chap e 4
4.4 Boo s ap app oxima ion
We p opose a boo s ap p ocedu e in o de o app oxima e he c i ical alues o he
es in a p ac ical si ua ion. The esampling p ocedu e is based on a smoo hed
e sion o he ‘nai e boo s ap’ desc ibed in E on (1981) and s udied in Ak i-
as (1986).
Fo i= 1, . . . , n, es ima e he censo ed esiduals in a non pa ame ic way
ˆ
Ei=Zi−ˆm(Xi)
ˆσ(Xi).
F om he censo ed sample o es ima ed esiduals ( ˆ
Ei,∆i), i= 1, . . . , n, compu e
he Kaplan-Meie es ima o ˆ
Fεand s anda dize hese esiduals in o de o e i y
he ini ial assump ion o ha ing loca ion unc ion 0 and scale unc ion 1 (i λ1=
Rˆ
F−1
ε(s)J(s)ds and λ2= (Rˆ
F−1
ε(s)2J(s)ds−λ2
1)1/2 hen he s anda dized esiduals
a e ˜
Ei= ( ˆ
Ei−λ1)/λ2).
The boo s ap p ocedu e we p opose consis s o he ollowing s eps. Fo ixed
Band o b= 1, . . . , B,
1. Fo i= 1, . . . , n:
•Le Y∗
i,b =mˆ
θ(Xi) + ˆσ(Xi)ε∗
i,b, whe e ε∗
i,b =Vi,b +aSi,b,Vi,b is d awn
om ˆ
Fε(s anda dized), Si,b is a andom a iable wi h mean ze o and
a iance one o in oduce a small pe u ba ion in he esiduals ( he
pe u ba ion is con olled by he cons an a). No e ha he boo s ap
esponses ollow he null hypo hesis by cons uc ion.
•Selec a andom C∗
i,b om a smoo hed e sion o ˆ
G(·|Xi), which is he
Be an es ima o o he condi ional dis ibu ion G(·|Xi) o he censo ing
a iable ob ained by eplacing ∆iby 1−∆iin he exp ession o ˆ
F(·|Xi).
•Le Z∗
i,b = min{Y∗
i,b, C∗
i,b}and ∆∗
i,b =I(Y∗
i,b ≤C∗
i,b).
2. The boo s ap sample is {(Xi, Z∗
i,b,∆∗
i,b), i = 1, . . . , n}.
3. Le T∗
KS,b and T∗
CM,b be he es s a is ics calcula ed wi h he boo s ap sam-
ple.
Goodness-o - i es s in censo ed eg ession 103
Le T∗
KS,(b)be he b- h o de s a is ic o T∗
KS,1, . . . , T∗
KS,B, and analogously o
T∗
CM,(b). Then T∗
KS,([(1−α)B]) and T∗
CM,([(1−α)B]) app oxima e he (1 −α)-quan iles o
he dis ibu ion o TKS and TCM unde he null hypo hesis espec i ely.
4.5 Some simula ions
We p esen some simula ion esul s in o de o s udy he ini e-sample beha io o
he goodness-o - i es and he boo s ap app oxima ion o he c i ical alues.
The eg ession and a iance unc ions a e hose co esponding o he choice
J(s) = 0.75−1I(0 ≤s≤0.75). We conside he ollowing eg ession unc ions:
(i)m(x) = x
(ii)m(x) = x+ 0.6(x−0.5)
(iii)m(x) = x+ 2x2
(i )m(x) = x+ 0.5 sin(4πx)
The a iance unc ion is σ2(x) = 0.5. The co a ia e is uni o mly dis ibu ed
in he in e al [0,1] and he e o is exponen ially dis ibu ed, ans o med such
ha R1
0F−1
ε(s)J(s)ds = 0 and R1
0F−1
ε(s)2J(s)ds = 1. The censo ing a iable
is C=m(X) + σ(X)ρ, whe e ρis independen o εand has su i al unc ion
1−Fρ(y) = (1 −Fε(y))β, wi h β= 1/3 (25% o censo ing) and β= 1 (50% o
censo ing).
We will es o wo di e en null hypo heses: a comple e linea model
H0:m(x) = θ1+θ2x
(in his case models (i) and (ii) co espond o he null hypo hesis and (iii) and (i )
co espond o he al e na i e hypo hesis) and a linea model h ough he o igin
H0:m(x) = θx
(in his case model (i) co esponds o he null hypo hesis and models (ii)-(i )
co espond o he al e na i e hypo hesis). The pa ame e is es ima ed by using he
me hod p oposed by Ak i as (1996) o polynomial models desc ibed in Sec ion 4.2.
104 Chap e 4
Table 4.1 shows he ejec ion p obabili ies in 1000 ials o sample sizes n= 100
and n= 200 and signi icance le els α= 0.05 and α= 0.10. We choose he
Epanechniko ke nel K(u) = 0.75(1 −u2)I(|u|<1) o calcula e he weigh s ha
appea in he Be an es ima o . The bandwid h was chosen o he o m h=cn−3/10
and he cases c= 0.75 and c= 1 a e displayed. In he boo s ap we use B= 200
eplica ions, a=n−3/10 and Si,b is a s anda d no mal.
The h eshold Tin he de ini ion o he es s a is ics was chosen o be he
la ges obse ed alue o he sample o censo ed esiduals es ima ed unde he null
hypo hesis. The le el is well app oxima ed in mos cases, al hough his app o-
xima ion ge s wo se when he da a a e hea ily censo ed (50% o censo ing). The
beha io o he powe is as expec ed: i inc eases wi h he sample size and i de-
c eases wi h he amoun o censo ed da a. Model (iii) is e y di icul o dis inguish
om a linea model, especially when he amoun o censo ed da a is 50%.
We belie e ha he choice o he h eshold Tmay ha e an impac on he powe .
When he da a a e hea ily censo ed, he Kaplan-Meie es ima o s ha e la ge jumps
in he igh ail o he dis ibu ion. This may p oduce la ge alues o he es
s a is ics e en unde he null hypo hesis. In Table 4.2 we ha e epea ed he same
simula ions by using as a h eshold he quan ile o o de ˆ
F−1
ε0(ˆ
Fε0(+∞)−0.10) o
a oid his p oblem. Now he le el beha es easonably well and he powe is be e
han in he p e ious able, especially when he amoun o censo ing is 50% and
he sample size is 200.
4.6 Da a analysis
We illus a e he p oposed goodness-o - i es on a da a se conce ning 90 male
pa ien s su e ing om la ynx cance , diagnosed and ea ed du ing he pe iod
1970-1978 in he Ne he lands. Mo e de ails abou his da a se can be ound
in Ka daun (1983). The a iable o in e es is he ime be ween i s ea men
and dea h. A he end o he s udy 40 pa ien s we e ali e ( hei su i al imes
a e censo ed). Heuchenne and Van Keilegom (2005) sugges ed a linea model o
explain he ela ionship be ween he log o he age o he pa ien a diagnosis as
Goodness-o - i es s in censo ed eg ession 105
Table 4.1: Rejec ion p obabili ies unde models (i) o (i ) o he es s based on TKS
and TCM . The h eshold Tis he la ges obse ed alue o he sample o censo ed
esiduals es ima ed unde he null hypo hesis.
c= 0.75 c= 1
TKS TCM TKS TCM
% Cens. nModel 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
H0:m(x) = θ1+θ2x
25% 100 (i) 0.054 0.098 0.049 0.097 0.043 0.086 0.042 0.096
(ii) 0.044 0.097 0.048 0.105 0.039 0.080 0.041 0.094
(iii) 0.124 0.202 0.130 0.220 0.099 0.165 0.122 0.210
(i ) 0.290 0.423 0.307 0.472 0.201 0.331 0.214 0.389
200 (i) 0.052 0.108 0.047 0.102 0.050 0.114 0.053 0.110
(ii) 0.044 0.110 0.047 0.106 0.038 0.096 0.051 0.106
(iii) 0.260 0.380 0.263 0.375 0.236 0.365 0.267 0.387
(i ) 0.727 0.838 0.760 0.877 0.604 0.761 0.723 0.848
50% 100 (i) 0.039 0.118 0.065 0.135 0.020 0.080 0.036 0.115
(ii) 0.038 0.103 0.055 0.136 0.030 0.086 0.042 0.116
(iii) 0.038 0.106 0.049 0.117 0.036 0.084 0.040 0.097
(i ) 0.066 0.174 0.070 0.194 0.050 0.124 0.046 0.126
200 (i) 0.027 0.110 0.055 0.125 0.021 0.070 0.045 0.118
(ii) 0.029 0.098 0.050 0.118 0.020 0.081 0.045 0.112
(iii) 0.040 0.108 0.063 0.137 0.023 0.087 0.045 0.114
(i ) 0.155 0.310 0.247 0.435 0.080 0.207 0.156 0.327
H0:m(x) = θx
25% 100 (i) 0.056 0.112 0.056 0.107 0.057 0.108 0.049 0.098
(ii) 0.311 0.427 0.356 0.496 0.283 0.382 0.336 0.466
(iii) 0.439 0.571 0.471 0.601 0.343 0.456 0.438 0.552
(i ) 0.256 0.398 0.148 0.293 0.225 0.385 0.132 0.279
200 (i) 0.065 0.118 0.071 0.121 0.061 0.114 0.060 0.117
(ii) 0.556 0.667 0.596 0.726 0.531 0.636 0.593 0.709
(iii) 0.749 0.827 0.743 0.824 0.688 0.765 0.732 0.807
(i ) 0.703 0.850 0.661 0.865 0.707 0.845 0.693 0.864
50% 100 (i) 0.041 0.103 0.060 0.152 0.030 0.078 0.056 0.117
(ii) 0.098 0.195 0.176 0.321 0.061 0.148 0.134 0.277
(iii) 0.130 0.242 0.215 0.391 0.061 0.154 0.176 0.331
(i ) 0.127 0.259 0.079 0.169 0.098 0.213 0.065 0.137
200 (i) 0.044 0.112 0.059 0.141 0.027 0.088 0.053 0.118
(ii) 0.166 0.295 0.335 0.499 0.135 0.248 0.307 0.471
(iii) 0.280 0.426 0.426 0.590 0.190 0.324 0.394 0.561
(i ) 0.327 0.527 0.356 0.564 0.274 0.457 0.359 0.551
106 Chap e 4
Table 4.2: Rejec ion p obabili ies unde models (i) o (i ) o he es s based on TKS
and TCM . The h eshold Tis he quan ile o o de ˆ
F−1
ε0(ˆ
Fε0(+∞)−0.10) o he
sample o censo ed esiduals es ima ed unde he null hypo hesis.
c= 0.75 c= 1
TKS TCM TKS TCM
% Cens. nModel 0.050 0.100 0.050 0.100 0.050 0.100 0.050 0.100
H0:m(x) = θ1+θ2x
25% 100 (i) 0.063 0.110 0.051 0.107 0.053 0.108 0.054 0.108
(ii) 0.056 0.112 0.052 0.116 0.052 0.092 0.046 0.100
(iii) 0.142 0.226 0.152 0.241 0.127 0.206 0.152 0.241
(i ) 0.340 0.491 0.378 0.542 0.263 0.411 0.290 0.466
200 (i) 0.062 0.128 0.055 0.114 0.061 0.131 0.061 0.115
(ii) 0.057 0.130 0.051 0.115 0.061 0.114 0.058 0.109
(iii) 0.302 0.410 0.277 0.385 0.301 0.431 0.309 0.438
(i ) 0.768 0.868 0.789 0.898 0.704 0.823 0.782 0.886
50% 100 (i) 0.032 0.060 0.050 0.111 0.019 0.044 0.037 0.112
(ii) 0.034 0.064 0.052 0.114 0.029 0.055 0.043 0.090
(iii) 0.064 0.116 0.089 0.146 0.051 0.090 0.059 0.126
(i ) 0.192 0.333 0.196 0.361 0.112 0.230 0.106 0.258
200 (i) 0.034 0.071 0.055 0.109 0.036 0.072 0.050 0.101
(ii) 0.034 0.075 0.054 0.112 0.032 0.059 0.042 0.097
(iii) 0.114 0.204 0.125 0.226 0.090 0.172 0.120 0.196
(i ) 0.500 0.669 0.611 0.757 0.392 0.561 0.526 0.670
H0:m(x) = θx
25% 100 (i) 0.062 0.119 0.059 0.108 0.063 0.119 0.049 0.104
(ii) 0.321 0.447 0.370 0.513 0.299 0.406 0.352 0.484
(iii) 0.462 0.592 0.496 0.617 0.365 0.479 0.455 0.576
(i ) 0.266 0.421 0.166 0.317 0.250 0.416 0.158 0.319
200 (i) 0.069 0.123 0.070 0.122 0.066 0.118 0.062 0.118
(ii) 0.559 0.680 0.606 0.734 0.547 0.644 0.606 0.722
(iii) 0.766 0.840 0.753 0.833 0.709 0.787 0.740 0.817
(i ) 0.713 0.869 0.677 0.877 0.735 0.858 0.714 0.887
50% 100 (i) 0.049 0.093 0.072 0.140 0.037 0.072 0.054 0.111
(ii) 0.185 0.274 0.298 0.418 0.154 0.230 0.262 0.392
(iii) 0.254 0.369 0.365 0.501 0.157 0.256 0.326 0.466
(i ) 0.228 0.373 0.142 0.254 0.185 0.326 0.111 0.196
200 (i) 0.060 0.098 0.067 0.128 0.043 0.084 0.051 0.112
(ii) 0.393 0.519 0.514 0.654 0.336 0.461 0.496 0.638
(iii) 0.550 0.670 0.628 0.749 0.451 0.580 0.603 0.719
(i ) 0.612 0.782 0.594 0.777 0.598 0.747 0.614 0.779

Goodness-o - i es s in censo ed eg ession 107
a co a ia e and he log o he su i al ime as esponse. These au ho s wo k wi h
he condi ional mean.
Figu e 4.1 shows he da a and eg ession cu es es ima ed nonpa ame ically
wi h sco e unc ions J(s) = 0.50−1I(0.25 ≤s≤0.75) and J(s) = 0.75−1I(0 ≤s≤
0.75). We belie e ha he choice o he sco e unc ion is no c ucial he e since he
da a seem o be homoscedas ic. The wo es ima ed cu es a e almos pa allel.
We ha e applied ou es o e i y he claimed linea model wi h bo h choices o
he unc ion J. The ob ained esul s we e e y simila , so we will only discuss he
esul s co esponding o J(s) = 0.50−1I(0.25 ≤s≤0.75). We ha e pe o med he
es o e a wide ange o bandwid hs ( om 0.15 o 0.35) and we ha e calcula ed
he p- alues based on 1000 boo s ap eplica ions. The h eshold Twas aken o
be he quan ile o o de ˆ
F−1
ε0(ˆ
Fε0(+∞)−0.10). The Kolmogo o -Smi no ype
s a is ic TKS p oduced p- alues be ween 0.29 and 0.99. On he o he hand, he
C am´e - on Mises ype s a is ic TCM ga e p- alues be ween 0.27 and 0.95. The
hypo hesis o linea i y can hen be clea ly accep ed.
Heuchenne and Van Keilegom (2005) also ga e boo s ap con idence in e als
o he pa ame e s o he linea eg ession. The in e al co esponding o he slope
o he eg ession line con ains ze o. Hence i is easonable o es o a cons an
model ins ead o he comple e linea model. We ha e applied he goodness-o - i
es o check he cons an model and he ob ained p- alues we e be ween 0.12 and
0.73 o TKS and be ween 0.08 and 0.80 o TCM . I seems ha he cons an model
can also be accep ed in his example.
All hese esul s a e summa ized in Figu e 4.2.
108 Chap e 4
log (age)
log (su i al ime)
3.6 3.8 4.0 4.2 4.4 4.6
-3 -2 -1 0 1 2 3
Figu e 4.1: Sca e plo o ‘log(su i al ime)’ e sus ‘log(age)’ (c osses o un-
censo ed da a and ci cles o censo ed da a) and es ima ed eg ession cu es wi h
J(s) = 0.50−1I(0.25 ≤s≤0.75) (solid line) and J(s) = 0.75−1I(0 ≤s≤0.75)
(dashed line).
h
p- alue
0.15 0.20 0.25 0.30 0.35
0.0 0.2 0.4 0.6 0.8 1.0
h
p- alue
0.15 0.20 0.25 0.30 0.35
0.0 0.2 0.4 0.6 0.8 1.0
Figu e 4.2: G aphs o he p- alues as unc ion o he bandwid h hwhen es ing
o a linea model (le ) and o a cons an model ( igh ) wi h he es s a is ics
TKS (line wi h ci cles) and TCM (line wi h c osses). The sco e unc ion is J(s) =
0.50−1I(0.25 ≤s≤0.75). The solid ho izon al line co esponds o a p- alue o
0.05.
Goodness-o - i es s in censo ed eg ession 109
4.7 P oo s
P oo o Theo em 4.1. The di ec implica ion is i ial. On he o he hand,
assume ha he e exis s a θ0such ha Fε0(y) = Fε(y). We can w i e
PµY−mθ0(X)
σ(X)≤y¶=PµY−m(X)
σ(X)+m(X)−mθ0(X)
σ(X)≤y¶,
o equi alen ly
Pµexp ½Y−mθ0(X)
σ(X)¾≤y¶
=Pµexp ½Y−m(X)
σ(X)¾exp ½m(X)−mθ0(X)
σ(X)¾≤y¶,
o all y. The esiduals (Y−m(X))/σ(X) and he co a ia e Xa e independen ,
hence he momen s o he dis ibu ions abo e e i y he ela ion
E"µexp ½Y−mθ0(X)
σ(X)¾¶2ν#
=E"µexp ½Y−m(X)
σ(X)¾¶2ν#E"µexp ½m(X)−mθ0(X)
σ(X)¾¶2ν#,
and hence
E"µexp ½m(X)−mθ0(X)
σ(X)¾¶2ν#= 1,
o all ν∈N. Ca leman’s condi ion (see e.g. Felle , 1966) ensu es ha
Pµexp ½m(X)−mθ0(X)
σ(X)¾= 1¶= 1
o
Pµm(X)−mθ0(X)
σ(X)= 0¶= 1.
This and he con inui y o mimplies he equali y o m(x) and mθ0(x) o all
x∈RX.
P oo o Theo em 4.2. Since we a e wo king unde he null hypo hesis, he e
exis s θ0such ha m=mθ0. F om he p oo o P oposi ion A.2 in Van Keilegom
110 Chap e 4
and Ak i as (1999), we ha e ha
ˆ
He0(y)−He0(y) =1
n
n
X
i=1
I(Ei0≤y)−He0(y) (4.6)
+Zhe0(y|x)mˆ
θ(x)−mθ0(x)
σ(x) X(x)dx
+Zyhe0(y|x)ˆσ(x)−σ(x)
σ(x) X(x)dx +oP(n−1/2),
uni o mly in −∞ < y ≤T. The las e m is oP(n−1/2) because o he uni o m
consis ency o mˆ
θand ˆσ. The consis ency o ˆσis gi en by P oposi ion 4.5 in Van
Keilegom and Ak i as (1999), and he consis ency o mˆ
θcan be ob ained in a
simila way.
De ine he class o unc ions MΘ(RX) = {x→(mθ(x)−m(x))/σ(x), θ ∈Θ}.
Fi s ly, his class e i ies P((mˆ
θ(x)−m(x))/σ(x)∈MΘ(RX)) →1 as n→ ∞
because o he consis ency o he pa ame e es ima e. Secondly, he b acke ing
numbe N[ ](λ2, MΘ(RX), L2(P)) = O(λ−2p) o any λ > 0 because o he compac -
ness o he pa ame ic space Θ. This b acke ing numbe is smalle han o he
class C1+δ
1(RX) de ined in Lemma A.1 in Van Keilegom and Ak i as (1999). Then
we can eplace he class C1+δ
1(RX) by he class MΘ(RX) in ha Lemma and his
jus i ies exp ession (4.6).
Using he exp ession (4.5), he i s in eg al in (4.6) can be w i en as
Zhe0(y|x)mˆ
θ(x)−mθ0(x)
σ(x) X(x)dx
=n−1
n
X
i=1 Zhe(y|u)σ−1(u)ϕθ0(u, Xi, Zi,∆i) X(u)du +oP(n−1/2).
F om P oposi ion 4.9 in Van Keilegom and Ak i as (1999) and a Taylo expan-
sion, he second in eg al in (4.6) becomes
Zyhe0(y|x)ˆσ(x)−σ(x)
σ(x) X(x)dx =−n−1
n
X
i=1
yhe0(y|Xi)ζ(Zi,∆i|Xi) + oP(n−1/2).
Re e ences 117
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Heuchenne, C. and Van Keilegom, I. (2005). Polynomial eg ession wi h censo ed
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by Ann. Ins . S a is . Ma h.
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118 Re e ences
Kulaseke a, K. B. (1995). Compa ison o eg ession cu es using quasi- esiduals.
J. Ame . S a is . Assoc. 90, 1085-1093.
Kulaseke a, K. B. and Wang, J. (1997). Smoo hing pa ame e selec ion o powe
op imali y in es ing o eg ession cu es. J. Ame . S a is . Assoc. 92, 500-511.
Kulaseke a, K. B. and Wang, J. (2001). A es o equali y o eg ession cu es
using Gˆa eaux sco es. Aus. N. Z. J. S a . 43, 89-99.
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unc ions: exac and asymp o ic heo y. Ann. S a is . 26, 2339-2368.
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Neumeye , N. and De e, H. (2005). A no e on one-sided nonpa ame ic analysis
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Pa do-Fe n´andez, J. C. (2005). Compa ison o e o dis ibu ions in nonpa ame ic
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Pa do-Fe n´andez, J. C. and Van Keilegom, I. (2005). Compa ison o eg ession
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Pa do-Fe n´andez, J. C., Van Keilegom, I. and Gonz´alez-Man eiga, W. (2005a).
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Re e ences 119
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Summa y in Galician /
Resumo en galego
A eg esi´on cons i ´ue un p oblema undamen al na es a ´ıs ica. Un modelo de
eg esi´on desc ibe a elaci´on en e unha a iable explica i a ou co a iable Xe
unha a iable espos a Y. Nun con ex o non pa am´e ico, a elaci´on en e YeX
p´odese exp esa como
Y=m(X) + σ(X)ε,
onde m´e unha unci´on de eg esi´on sua e, σ´e a unci´on de a ianza, que ep esen a
posible he e ocedas icidade, e ε´e o e o do modelo de eg esi´on, que asumimos
independen e da co a iable X. Obse a emos da os do ec o alea o io (X, Y ).
Se a a iable espos a Y ep esen a un empo, con ecuencia oco e que os
da os es ´an incomple os debido a di e sas az´ons. Unha on e impo an e de in-
comple i ude ´e a censu a. Is o signi ica que exis e unha a iable alea o ia C,
chamada a iable de censu a, que pode ocul a a nosa a iable de in e ese po que
s´o podemos obse a Ycando o seu alo ´e meno ou igual ca C. Dun xei o m´ais
o mal e nun con ex o condicional, obse amos o ec o alea o io (X, Z, ∆), onde
Z= min{Y, C}´e o empo obse ado asociado ´a co a iable Xe ∆ = I(Y≤C)
´e o indicado de censu a (1 ep esen a un da o non censu ado e 0 ep esen a un
da o censu ado). Asumimos que a co a iable non es ´a censu ada e que as a iables
espos a Ye de censu a Cson independen es dada a co a iable X.
Habi ualmen e as unci´ons de eg esi´on e de a ianza son a media e a a ianza
condicionais espec i amen e, pe o am´en se poden conside a ou os uncionais.
121
122 Resumo en galego
Se a unci´on de eg esi´on ´e a media condicional en ´on
m(x) = ZydF(y|x),
onde F(·|x) ´e a unci´on de dis ibuci´on condicional da a iable espos a Ydado o
alo xda co a iable X. Es a ´ul ima exp esi´on am´en se pode esc ibi como
m(x) = Z1
0
F−1(s|x)ds
onde F−1(s|x) = in { ;F( |x)≥s}´e a co esponden e unci´on cuan il condicional.
Pa a es ima mabonda con dispo˜ne dun es imado da unci´on de dis ibuci´on
condicional e inco po alo ´a exp esi´on an e io . Cando a a iable espos a ´e cen-
su ada pode oco e que as colas de ei as do es imado da unci´on de dis ibuci´on
condicional deixen de se consis en es debido ao p opio mecanismo que xe a a cen-
su a. Nes e caso esul a ´u il ede ini as unci´ons de eg esi´on e de a ianza do
modelo de eg esi´on como
m(x) = Z1
0
F−1(s|x)J(s)ds
e
σ2(x) = Z1
0
F−1(s|x)2J(s)ds −m2(x),
onde J´e unha unci´on al que R1
0J(s)ds = 1. A elecci´on da unci´on Jcond´ucenos
a dis in as unci´ons condicionais de localizaci´on e de escala (que am´en denomi-
namos, nun senso m´ais amplo, unci´ons de eg esi´on e de a ianza). En pa icula ,
se J(s) = I(0 ≤s≤1) ob emos de no o a media e a ianza condicionais. Unha
elecci´on in e esan e ´e J(s) = (q−p)−1I(p≤s≤q), que co esponde a medias e a-
ianzas eco adas. A mediana condicional ou ou os cuan ´ıs condicionais p´odense
e como l´ımi es de medias eco adas.
O obxec i o p incipal des a ese consis e en desen ol e con as es de hip´o eses
sob e a unci´on de eg esi´on men di e sos ´ambi os, an o pa a da os comple os
coma pa a da os censu ados. Os con as es p opos os es ´an baseados na unci´on
de dis ibuci´on dos e os do modelo de eg esi´on, que en dada po
Fε(y) = P(ε≤y) = PµY−m(X)
σ(X)≤y¶.

Summa y in Galician 123
A es imaci´on des a unci´on de dis ibuci´on implica a es imaci´on da unci´on de
eg esi´on e da unci´on de a ianza condicional.
No Cap´ı ulo 1 acemos unha b e e in oduci´on sob e os no os m´e odos desen-
ol idos nes e aballo e dos esul ados ob idos. Realizamos am´en unha e isi´on
bibliog ´a ica sob e compa aci´on de cu as de eg esi´on e sob e con as es de bon-
dade de axus e pa a modelos pa am´e icos de eg esi´on. Finalmen e inclu´ımos un
b e e epaso das ´ecnicas de es imaci´on non pa am´e ica na eg esi´on, an o con
da os comple os coma con da os censu ados.
No Cap´ı ulo 2 p opo˜nemos un no o m´e odo de compa aci´on de cu as de
eg esi´on desde un pun o de is a o almen e non pa am´e ico. A compa aci´on
de dis in os g upos de indi iduos ou poboaci´ons ´e un p oblema com´un na es-
a ´ıs ica. Nun con ex o incondicional (sen p esenza de co a iables), p ocedemen-
os xa cl´asicos como o - es ou a An´alise da Co a ianza es ´an elacionados con
es e p oblema. A exis encia de co a iables asociadas ´a a iable de in e ese le a o
p oblema a un con ex o condicional. Ago a o obxec i o consis e en compa a as
co esponden es cu as de eg esi´on pa a sabe se o e ec o das co a iables sob e
as a iables espos a ´e o mesmo nos dis in os g upos de indi iduos. Se as cu as
de eg esi´on es ´an con idas nun modelo pa am´e ico en ´on o p oblema ed´ucese ´a
compa aci´on dos co esponden es pa ´ame os. Sen emba go, en moi as aplicaci´ons
p ´ac icas non con ´en asumi ning´un modelo pa am´e ico e a´ında as´ı a compa aci´on
das cu as de eg esi´on segue a se un p oblema de in e ese.
O p oblema de compa aci´on de cu as de eg esi´on oi a ado amplamen e
na li e a u a ao longo dos ´ul imos quince anos, pe o p incipalmen e en si uaci´ons
m´ais es ic i as c´a conside ada nes a ese. A maio ´ıa das e e encias exis en es
es ´an dedicadas a con as a a igualdade de soamen e d´uas cu as de eg esi´on
e as s´uas ex ensi´ons pa a compa a m´ais de d´uas cu as non son doadas. En
aplicaci´ons p ´ac icas o p oblema da compa aci´on de m´ais de d´uas cu as pode
apa ece de o ma na u al. Se a compa aci´on se ixese pa a pa en ´on am´en se
debe ´ıa aplica unha co ecci´on no ni el e como consecuencia a po encia pode e se
a ec ada. Is o mo i a a implemen aci´on de p ocedemen os xe ais pa a con as a a
124 Resumo en galego
igualdade de m´ais de d´uas cu as de eg esi´on, al e como acemos nes e aballo.
O modelo es a ´ıs ico conside ado nes a ese ´e o que se desc ibe a con inuaci´on.
Sexan k ec o es alea o ios independen es (Xj, Yj) que sa is an os seguin es mo-
delos de eg esi´on non pa am´e icos, pa a j= 1, . . . , k,
Yj=mj(Xj) + σj(Xj)εj,
onde as co a iables e˜nen sopo e com´un, e o e o εj´e independen e de Xje en
unci´on de dis ibuci´on Fεj. Sexa (Xij, Yij), i= 1, ..., nj, unha mos a de obse -
aci´ons independen es e iden icamen e dis ibu´ıdas do ec o alea o io (Xj, Yj),
pa a j= 1, . . . , k. Es amos in e esados en con as a a hip´o ese nula de igualdade
das unci´ons de eg esi´on
H0:m1=m2=··· =mk
on e ´a al e na i a xe al
Ha:mi6=mjpa a alg´un i, j ∈ {1,2, . . . , k}.
A idea do p ocedemen o de con as e p opos o e es udado nes a ese consis e en
compa a en cada poboaci´on a dis ibuci´on emp´ı ica dos esiduos coa dis ibuci´on
emp´ı ica dos esiduos es imados supo˜nendo que a hip´o ese nula ´e ce a. Fixe-
mos unha poboaci´on, digamos j. Sexan ˆmje ˆσjes imado es non pa am´e icos
da unci´on de eg esi´on mje da unci´on de a ianza σj(cons u´ıdos coa mos a
co esponden e ´a poboaci´on j), e sexa ˆmun es imado da unci´on de eg esi´on
com´un baixo a hip´o ese nula, que deno amos po m(cons u´ıdo u ilizando odas
as mos as de manei a conxun a). En ´on a can idade (Yij −ˆmj(Xij))/ˆσj(Xij)
es ima ´a o e o εij e a can idade (Yij −ˆm(Xij))/ˆσj(Xij) es ima ´a o mesmo alo
cando se aume que a hip´o ese nula ´e e dadei a. Cons u´ımos as dis ibuci´ons
emp´ı icas co esponden es a es es esiduos es imados
ˆ
Fεj(y) = 1
nj
nj
X
i=1
IµYij −ˆmj(Xij)
ˆσj(Xij)≤y¶,
e
ˆ
Fεj0(y) = 1
nj
nj
X
i=1
IµYij −ˆm(Xij)
ˆσj(Xij)≤y¶.
Summa y in Galician 125
Cando a hip´o ese nula de igualdade de cu as de eg esi´on ´e e dadei a, as d´uas
unci´ons de dis ibuci´on emp´ı icas an e io es es iman a co esponden e unci´on de
dis ibuci´on dos e os Fεjna poboaci´on j. Sen emba go, se a hip´o ese nula non ´e
ce a, es as unci´ons emp´ı icas es iman dis ibuci´ons di e en es po que os esiduos
es ´an medidos con espec o a cu as dis in as (a e dadei a unci´on de eg esi´on
na poboaci´on je a unci´on de eg esi´on com´un baixo a hip´o ese nula). Polo an o
calque a di e enza en e es as d´uas unci´ons de dis ibuci´on emp´ı icas d´a e idencia
en con a da hip´o ese nula de igualdade das cu as de eg esi´on. P´odese demos a
que a igualdade das cu as de eg esi´on queda ca ac e izada pola igualdade das
unci´ons de dis ibuci´on que se es iman coas dis ibuci´ons emp´ı icas an e io es.
A compa aci´on l´e ase a cabo a a ´es de es a ´ıs icos de con as e de ipo
Kolmogo o -Smi no e C am´e - on Mises de inidos sob e o p oceso emp´ı ico k-
dimensional
ˆ
W(y) = ( ˆ
W1(y), . . . , ˆ
Wk(y)) ,
onde ˆ
Wj(y) = n1/2
j(ˆ
Fεj0(y)−ˆ
Fεj(y)) pa a j= 1, . . . , k. Os esul ados e´o icos
ob idos incl´uen unha ep esen aci´on case segu a pa a a di e enza en e os dous
es imado es da dis ibuci´on dos e os ˆ
Fεj0(y)−ˆ
Fεj(y), a con e xencia eble do
p oceso mul idimensional ˆ
W(y) e a con e xencia en dis ibuci´on dos es a ´ıs icos
de con as e. Ademais am´en p obamos que es e p ocedemen o de con as e pode
de ec a al e na i as locais con e xendo ´a hip´o ese nula a unha axa n−1/2, ´ıpica
en p ocedemen os pa am´e icos. Nas demos aci´ons des es esul ados asin ´o icos
emp ´egase, ademais de ´ecnicas com´uns en es a ´ıs ica non pa am´e ica, a eo ´ıa
dos b acke ing numbe s e a e amen a de C am´e -Wold.
As dis ibuci´ons asin ´o icas dos es a ´ıs icos de con as e esul an se bas an e
complicadas. Como soluci´on pa a ap oxima os alo es c ´ı icos do con as e p o-
po˜nemos un mecanismo de emos axe boo s ap. A ep esen aci´on case segu a
ob ida pa a a di e enza das dis ibuci´ons emp´ı icas dos esiduos incl´ue na s´ua ex-
p esi´on a densidade dos e os do modelo de eg esi´on. Is o suxi e que se debe em-
p ega un m´e odo de emos axe baseado nos esiduos sua izados. Na cons ucci´on
das mos as boo s ap u il´ızase a unci´on de eg esi´on es imada de manei a con-
xun a con odas as obse aci´ons. Des a manei a as a iables espos a e i ica ´an a
126 Resumo en galego
hip´o ese nula e os cuan ´ıs dos es a ´ıs icos de con as e ob idos coas mos as boo -
s ap ap oxima ´an os co esponden es cuan ´ıs das dis ibuci´ons dos es a ´ıs icos de
con as e baixo a hip´o ese nula.
Nun es udo de simulaci´on amosamos o compa amen o p ´ac ico do m´e odo de
con as e e da ap oximaci´on boo s ap dos alo es c ´ı icos. Conside amos esce-
na ios de simulaci´on con d´uas e es cu as de eg esi´on. Os esul ados ob idos
amosan un compo amen o sa is ac o io na maio ´ıa dos casos conside ados. A
ap oximaci´on do ni el ´e boa e a po encia mello a cando aumen a o ama˜no das
mos as. Ademais obs´e ase que a elecci´on do pa ´ame o de sua izado que se
p ecisa pa a a es imaci´on dos esiduos (na es imaci´on das unci´ons de eg esi´on e
de a ianza) non en unha epe cusi´on impo an e nas p obabilidades de exei a-
men o.
Pa a ilus a a me odolox´ıa p opos a nes e cap´ı ulo ealizamos unha aplicaci´on
a da os eais ex a´ıdos do a qui o de da os do Jou nal o Applied Econome ics.
Consid´e anse a elaci´on en e o gas o o al e o gas o en comida dun conxun o
de amilias holandesas. Como co a iable omamos o loga i mo do gas o o al
mensual e como a iable espos a conside amos o loga i mo do gas o mensual
en comida. Realizamos o con as e de igualdade das co esponden es cu as de
eg esi´on dis inguindo as amilias polo n´ume o de pe soas que as compo˜nen.
Tam´en no Cap´ı ulo 2 se es uda un m´e odo pa a compa a as dis ibuci´ons dos
e os dos modelos de eg esi´on en dis in os g upos. Es a ´e unha suposici´on ela i-
amen e com´un nalg´uns m´e odos de compa aci´on de cu as de eg esi´on exis en es
na li e a u a, a´ında que non ´e necesa io pa a o noso. Sexa Fεja dis ibuci´on do
e o εjna poboaci´on j. Ago a que emos con as a a hip´o ese nula de igualdade
das unci´ons de dis ibuci´on dos e os
H0ε:Fε1=Fε2=. . . =Fεk
on e ´a al e na i a xe al de que exis e algunha di e enza en e es as dis ibuci´ons.
Deno amos po Fεa unci´on de dis ibuci´on com´un cando se e i ica a hip´o ese