Testing for changes in the error distribution in functional linear models

Neumeye , Na alie; Selk, Leonie
A icle — Published Ve sion
Tes ing o changes in he e o dis ibu ion in unc ional
linea models
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REGULAR ARTICLE
Tes ing o changes in he e o dis ibu ion in unc ional
linea models
Na alie Neumeye 1·Leonie Selk1
Recei ed: 3 Ap il 2024 / Re ised: 6 No embe 2024
© The Au ho (s) 2025
Abs ac
We conside linea models wi h scala esponses and co a ia es om a sepa able
Hilbe space. The aim is o de ec change poin s in he e o dis ibu ion, based on
sequen ial esidual empi ical dis ibu ion unc ions. Expansions o hose es ima ed
unc ions a e mo e challenging in models wi h in ini e-dimensional co a ia es han
in eg ession models wi h scala o ec o - alued co a ia es due o a slowe a e
o con e gence o he pa ame e es ima o s. Ye he sugges ed change poin es is
asymp o ically dis ibu ion- ee and consis en o one-change poin al e na i es. In
he la e case we also show consis ency o a change poin es ima o .
Keywo ds Change-poin s ·Func ional da a analysis ·Regula ized unc ion
es ima o s ·Reg ession ·Residual p ocesses
Ma hema ics Subjec Classi ica ion P ima y 62R10; Seconda y 62G10 ·62G30
1 In oduc ion
We conside a unc ional linea model Y=α+X,β+εwi h scala esponse Y
and co a ia es X om a sepa able Hilbe space, e.g. L2([0,1]). S uc u al changes
in he dis ibu ion can appea , e en when he pa ame e s αand βdo no change.
Fo his eason we ocus on de ec ing changes in he e o dis ibu ion. I he e o s
we e obse able one could use he classical es (and change poin es ima o s) based
on he di e ence o he sequen ial empi ical dis ibu ion unc ions o he i s n 
and he las n−n e o e ms om a sample o nobse a ions, see Csö gö e al.
(1997), Pica d (1985), Ca ls ein (1988), Dümbgen (1991), Ha iz e al. (2005) and
Ha iz e al. (2007). In a eg ession model hose es s ha e o be based on es ima ed
esiduals ˆε=Y−ˆα−X,ˆ
β. Simila es s ha e been conside ed by Bai (1994)in
BNa alie Neumeye
na alie.neumeye @uni-hambu g.de
Leonie Selk
leonie.selk@uni-hambu g.de
1Fachbe eich Ma hema ik, Uni e si ä Hambu g, Hambu g, Ge many
0123456789().: V,- ol 123
33 Page 2 o 17 N. Neumeye , L. Selk
he con ex o ARMA-models, by Koul (1996) in he con ex o nonlinea ime se ies,
by Ling (1998) o nons a iona y au o eg essi e models, and by Neumeye and Van
Keilegom (2009) and Selk and Neumeye (2013) o nonpa ame ic independen and
ime se ies eg ession models. Typically he asymp o ic dis ibu ion is de i ed using
asymp o ic expansions o esidual-based empi ical dis ibu ion unc ions. Fo models
wi h unc ional co a ia es hose expansions can be p oblema ic because inne p oduc s
X,ˆ
β−βappea and hose can ha e a slow a e o con e gence [see Ca do e al.
(2007), Shang and Cheng (2015), Yeon e al. (2023)]. Howe e , we show ha unde
e y simple non- es ic i e assump ions hose e ms cancel o he sugges ed change
poin es s a is ic and hus he asymp o ic dis ibu ion is he same as based on ue
(unobse ed) e o s.
Change poin es ing and es ima ion o unc ional da a, and o he pa ame e in
unc ional linea models ha e been conside ed in he li e a u e, bu no o he e o
dis ibu ion. Tes s o changes in he unc ional mean and in he pa ame e unc ion o
au o eg essi e models a e conside ed in chap e s 6 and 14 in Ho á h and Kokoszka
(2012). Be kes e al. (2009) p opose a CUSUM es ing p ocedu e o de ec a change in
he mean o unc ional obse a ions. They apply p ojec ions on p incipal componen s
o he da a o es ima e he mean. Aue e al. (2009) ex end his esul and in oduce an
es ima o o he change poin in his model and de i e i s limi dis ibu ion. As on and
Ki ch (2012) conside he same ype o model wi h epidemic changes and dependen
da a. Aue e al. (2018) conside how o de ec and da e s uc u al b eaks in he mean
o unc ional obse a ions wi hou he applica ion o dimension educ ion echniques
(as unc ional p incipal componen analysis). Aue e al. (2014) p opose a moni o ing
p ocedu e o de ec s uc u al changes in unc ional linea models wi h unc ional
esponse, allowing o dependence in he da a, including unc ional au o eg essi e
p ocesses. They es o a change in he eg ession ope a o , which is he analogue o
ou β, based on unc ional p incipal componen analysis. A linea eg ession model
wi h scala esponse is conside ed in Ho á h e al. (2024) who p opose a es s o
he de ec ion o mul iple change poin s in he eg ession pa ame e . The eg esso s in
hei model can be unc ional and can include lagged alues o he esponse.
The pape is o ganized as ollows. In Sec . 2we de ine he es s a is ic and p esen
model assump ions o ob ain he asymp o ically dis ibu ion- ee hypo hesis es . In
Sec . 3we discuss he assump ions on he pa ame e es ima o s and some examples.
In Sec . 4consis ency o he es as well as o a change poin es ima o is conside ed
in he con ex o one change poin . Fini e sample p ope ies a e shown in Sec . 5.
Sec ion6concludes he pape , in pa icula wi h an ou look on goodness-o - i es ing.
The p oo s a e gi en in he appendix.
2 Model, es s a is ic and main esul unde he null
Le Hbe a sepa able Hilbe space wi h inne p oduc ·,·, co esponding no m ·
and Bo el-sigma ield. Le (Xi,Yi),i=1,...,n, be an independen sample o (H×
R)- alued andom a iables de ined on he same p obabili y space wi h p obabili y
123
Tes ing o changes in he e o dis ibu ion… Page 3 o 17 33
measu e P. The da a a e modeled as unc ional linea model
Yi=α+Xi,β+εi,i=1,...,n,
wi h scala esponse Yiand H- alued co a ia e Xi, and wi h pa ame e s α∈R,βinH.
The co a ia es X1,...,Xna e assumed o be iid wi h EXi<∞, and he e o s
ε1,...,ε
na e independen , cen e ed, and independen o he co a ia es. Ou aim is
o es o change-poin s in he e o dis ibu ion. In his sec ion we conside he es
s a is ic unde he null hypo hesis, whe e he e o s a e iden ically dis ibu ed.
Le ˆαand ˆ
βdeno e es ima o s o he pa ame e s α∈Rand β∈H. We build
esiduals ˆεi=Yi−ˆα−Xi,ˆ
β,i=1...,n. The es s a is ic
Tn=sup
∈[0,1]
sup
z∈R|ˆ
Gn( ,z)|
based on he p ocess
ˆ
Gn( ,z)=n (n−n )
n3/2ˆ
Fn (z)−˜
Fn (z),
compa es o each k=1,...,n−1 he empi ical dis ibu ion unc ions
ˆ
Fk(z)=1
k
k

i=1
I{ˆεi≤z},˜
Fk(z)=1
n−k
n

i=k+1
I{ˆεi≤z}
o he i s kand las n−k esiduals, espec i ely. No e ha one can w i e
ˆ
Gn( ,z)=n 
n1/2ˆ
Fn (z)−ˆ
Fn(z).
Fo he asymp o ic dis ibu ion o he es s a is ic unde he null hypo hesis we
assume he ollowing condi ions. Le Pdeno e he dis ibu ion o (X1,ε
1).
(a.1) |ˆα−α|=oP(1),ˆ
β−β=oP(1)
(a.2) Le ε1,...,ε
nbe independen and iden ically dis ibu ed wi h cd F ha is
Hölde -con inuous o o de γ∈(0,1]wi h Hölde -cons an c.
(a.3) Pˆ
β−β∈B→1asn→∞ o a class B⊂Hsuch ha he unc ion class
F={(x,e)→ I{e≤ +x,b} | ∈R,b∈B}
is P-Donske .
Rema k 2.1 The assump ions a e e y mild and in pa icula less es ic i e han ypical
assump ions o asymp o ic dis ibu ion o esidual-based empi ical p ocesses, e en
o ini e-dimensional co a ia es. In assump ion (a.1) only consis ency is needed, no
a es o con e gence. Typically in he li e a u e abou esidual-based p ocedu es a
bounded e o densi y is assumed, see e.g. Ak i as and Van Keilegom (2001). Then
123
33 Page 4 o 17 N. Neumeye , L. Selk
(a.2) is ul illed o γ=1, bu (a.2) is less es ic i e in he cases γ∈(0,1). Sui able
condi ions o he gene al assump ion (a.3) a e discussed in Sec .3. One possibili y
o H=L2([0,1])is o assume smoo hness o βwhich is a ypical assump ion. I
γ∈(1
2,1]in assump ion (a.2), and βis in a Sobole -space wi h hi d de i a i es,
(a.2) holds o he es ima o ˆ
β om Yuan and Cai (2010). This es ima o can also be
applied o smalle γin (a.2) i highe smoo hness o βis assumed.
De ine he p ocess Gnas ˆ
Gn, bu based on he ue e o s ins ead o esiduals, i.e.
Gn( ,z)=n 
n1/2Fn (z)−Fn(z)
wi h
Fn (z)=1
n 
n 

i=1
I{εi≤z}.(2.1)
Fu he le Gbe a comple ely ucked B ownian shee , i.e. a cen e ed Gaussian p ocess
on [0,1]2wi h co a iance s uc u e
Co (G(s,u), G( , ))=(s∧ −s )(u∧ −u ).
Theo em 2.2 Unde he assump ions (a.1)–(a.3),
sup
∈[0,1]
sup
z∈R|ˆ
Gn( ,z)−Gn( ,z)|=oP(1), (2.2)
and hus he p ocess (ˆ
Gn( ,z)) ∈[0,1],z∈Rcon e ges weakly o (G( ,F(z))) ∈[0,1],z∈R.
The p oo o (2.2) in he heo em is gi en in he appendix. The weak con e gence o Gn
is a classical esul , see Bickel and Wichu a (1971), Sho ack and Wellne (1986). Wi h
he con inuous mapping heo em one ob ains he asymp o ic dis ibu ion o he es
s a is ic Tnunde he null hypo hesis o no change-poin , which is he dis ibu ion o
T=sup ,u∈[0,1]|G( ,u)|because Fis con inuous. The es s a is ic is asymp o ically
dis ibu ion- ee wi h he same limi dis ibu ion as o co esponding changepoin
es s based on iid obse a ions (no esiduals). Le ¯α∈(0,1)and qbe he (1−¯α)-
quan ile o T. Then he es ha ejec s he null hypo hesis i Tn>qhas asymp o ic
le el ¯α. Consis ency is conside ed in Sec .4.
Rema k 2.3 The choice o Tnas a Kolmogo o –Smi no ype es s a is ic is no
manda o y. In p inciple, any con inuous unc ional o he p ocess ˆ
Gncan be con-
side ed. The mos common ones, besides Tn, a e o C amé - on–Mises ype, e. g.
Tn,2=sup ∈[0,1]R|ˆ
Gn( ,z)|2dF(z)o Tn,3=1
0R|ˆ
Gn( ,z)|2dF(z)d .The
asymp o ic dis ibu ion o hese es s a is ics unde he null hypo hesis also ollows
om Theo em 2.2 and wi h
R|ˆ
Gn( ,z)|2dF(z)→R|G( ,F(z))|2dF(z)=1
0|G( ,x)|2dx,
123

Tes ing o changes in he e o dis ibu ion… Page 5 o 17 33
and hus hese es s a is ics a e asymp o ically dis ibu ion- ee as well. Howe e ,
Tn,2and Tn,3con ain he unknown quan i y Fand mus he e o e be modi ied in
o de o be applied. This can be done by eplacing he in eg al wi h he sample mean:
˜
Tn,2=sup ∈[0,1]1
nn
i=1|ˆ
Gn( ,ˆεi)|2and ˜
Tn,3=1
0
1
nn
i=1|ˆ
Gn( ,ˆεi)|2d .
3 Discussion o assump ions and examples
To show alidi y o he Donske -class assump ion (a.3) he e a e su icien condi ions
on co e ing numbe s o b acke ing numbe s. We discuss some speci ic condi ions on
he class B, examples o Hilbe spaces H, and es ima o s o he pa ame e unc ion
β ha ul ill he condi ions.
3.1 VC-class condi ion
Assump ion (a.3) can be de i ed om a VC- unc ion class condi ion o mula ed as
ollows. Assume ha Pˆ
β−β∈B→1asn→∞ o a class B⊂Hsuch ha he
class o maps
{H→R,x→x,b+ |b∈B, ∈R}(3.1)
is a VC-subg aph class. By de ini ion hen
{{(x,e)∈H×R|e≤x,b+ }|b∈B, ∈R}
is a VC-class o se s. The class F om (a.3) is he class o he co esponding indica o
unc ions and (a.3) is ul illed by Theo ems 8.19 and 9.2 in Koso ok (2008).
Example 3.1 We conside he Hilbe space H=L2([0,1])wi h inne p oduc
g,h=1
0g( )h( )d and no m g=(1
0g2( )d )1/2. Fo he pa ame e unc ion
βwe assume spa si y as in Lee and Pa k (2012). Le (φj)j∈Nbe a basis o Hand
assume β=j∈Jβjφj o some ini e, bu unknown index se J. Lee and Pa k
(2012) conside he es ima o ˆ
β=k
j=1ˆ
βjφjwi h
(ˆ
β1,..., ˆ
βk)=a g min
b1,...,bk∈R⎛
⎝1
n
n

i=1Yi−Yn−
k

j=1
bjXi−Xn,φj2+
k

j=1ˆwj|bj|⎞
⎠
2
,
whe e kis a chosen dimension-cu -o , ˆwja e sui able weigh s based on ini ial es i-
ma o s, and Yn=1
nn
i=1Yi,Xn=1
nn
i=1Xi. Fu he , ˆα=Yn−ˆ
β, Xn. Unde
sui able assump ions, in pa icula EX2<∞, and kis la ge han he la ges index
in J, Lee and Pa k (2012) show in hei Theo em 2 ha P(ˆ
βj=0 o j/∈J)→1 o
123
33 Page 6 o 17 N. Neumeye , L. Selk
n→∞. Thus we can se
B=⎧
⎨
⎩
j∈J
bjφjbj∈R∀j∈J⎫
⎬
⎭
and ob ain P(ˆ
β−β∈B)→1 o n→∞. Fu he , he class o maps in (3.1), i.e.
⎧
⎨
⎩
H→R,x→ 
j∈J
bjx,φj+ bj∈R∀j∈J, ∈R⎫
⎬
⎭
is a ini e dimensional ec o space and hus a VC-class, see Lemma 2.6.15 in an
de Vaa and Wellne (1996). Then as discussed abo e alidi y o (a.3) ollows. Fu -
he mo e, om Theo em 2 in Lee and Pa k (2012) i also ollows ha ou assump ion
(a.1) is ul illed, and hus unde assump ion (a.2) he asse ion o Theo em 2.2 holds.
3.2 B acke ing numbe condi ion
In his subsec ion we assume ha His a sepa able Hilbe space o eal- alued unc-
ions (o ec o s wi h eal componen s) and he inne p oduc is inc easing in he sense
ha om h≤g(poin wise o unc ions; componen wise o ec o s) i ollows ha
h,x≤g,x o all x∈Hwi h x≥0. Then assump ion (a.3) can be eplaced by
he condi ion in he nex lemma.
Lemma 3.2 Assume (a.1),(a.2) and Pˆ
β−β∈B→1as n →∞ o a unc ion
class B⊂Hsuch ha he b acke ing numbe ul ills log N[](B,,·)≤K/1/k o
some k >1/γ .He eγis he Hölde -o de om assump ion (a.2). Then assump ion
(a.3) holds.
The p oo is gi en in he appendix.
Example 3.3 We conside he Hilbe space H=L2([0,1])wi h inne p oduc
g,h=1
0g( )h( )d and no m g=(1
0g2( )d )1/2. We assume β∈
Wm
2([0,1]) o some m>2 and he Sobole -space
Wm
2([0,1])=b:[0,1]→R|b(j)is absolu ely con inuous o j=0,...,m−1,
and b(m)<∞,
whe e b(0)=b, and b(j)deno es he j- h de i a i e o b,j≥1. We conside he
egula ized es ima o s in Yuan and Cai (2010), i.e.
ˆα, ˆ
β=a g min
a∈R,b∈Wm
2([0,1])1
n
n

i=1Yi−a+Xi,b2+λn
b(m)
2
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Tes ing o changes in he e o dis ibu ion… Page 7 o 17 33
o a sui able posi i e sequence λncon e ging o ze o. Con e gence a es o ˆ
βand
i s de i a i es can be ound in Co olla ies 10 and 11 in Yuan and Cai (2010). Unde
sui able assump ions one ob ains 
ˆ
β(j)−β(j)
=oP(1) o j=0,1,2, and hus
P(ˆ
β−β∈B)→1 o he unc ion class
B=b∈W2
2[0,1]:b+b(2)≤1.
By Co olla y 4.3.38 in Giné and Nickl (2021) and Lemma 9.21 in Koso ok (2008) he
class B ul ills he b acke ing numbe condi ion in Lemma 3.2 o k=2. Thus he
assump ions (a.1)–(a.3) a e ul illed i Fis Hölde -con inuous o o de γ∈(1
2,1].Less
es ic i e assump ions on F, i.e. γ≤1
2, equi e o his concep highe smoo hness
o β.
4 Fixed one-change poin al e na i e: consis ency o he es and
change poin es ima o
In his sec ion we conside ixed al e na i es wi h one change poin a index k∗
n=
nϑ∗wi h ϑ∗∈(0,1). We w i e he unc ional linea model as in Sec .2unde he
ollowing assump ion.
(a.2)’ Assume ε1,...,ε
k∗
na e iid wi h cd F1, and εk∗
n+1,...,ε
na e iid wi h cd
F2= F1.Le F1and F2be Hölde -con inuous o o de γ1,γ
2∈(0,1]wi h
Hölde -cons an c1,c2, espec i ely.
Le u he P1deno e he dis ibu ion o (X1,ε
1)(be o e he change) and P2deno e
he dis ibu ion o (Xn,ε
n)(a e he change). Fo he empi ical dis ibu ion unc ions
ˆ
Fkand ˜
Fkas in Sec . 2we ob ain he ollowing asymp o ic esul .
Lemma 4.1 Unde assump ions (a.1) and (a.2)’ and i (a.3) is alid o P =P1and
P=P2, i holds ha
sup
z∈R|ˆ
Fk∗
n(z)−F1(z)|=oP(1)and sup
z∈R|˜
Fk∗
n(z)−F2(z)|=oP(1).
The p oo is gi en in he appendix. Now no e ha
Tn
n1/2≥k∗
n(n−k∗
n)
n2sup
z∈Rˆ
Fk∗
n(z)−˜
Fk∗
n(z),
and by Lemma 4.1 he igh hand side con e ges in p obabili y o he posi i e cons an
ϑ∗(1−ϑ∗)sup
z∈R|F1(z)−F2(z)|.
F om his i ollows ha es s ha ejec he null hypo hesis o no change-poin i
Tn>q o some q>0 (see Sec . 2) a e consis en .
123
33 Page 8 o 17 N. Neumeye , L. Selk
The es ima o o he change poin ϑ∗is based on he p ocess ˆ
Gnand is de ined as
ˆ
ϑn=min  :sup
z∈R|ˆ
Gn( ,z)|= sup
∈[0,1]
sup
z∈R|ˆ
Gn( ,z)|.
Lemma 4.2 Unde assump ions (a.1),(a.2)’ and i (a.3) holds o P =P1and P =P2,
he change poin es ima o is consis en , i.e.
|ˆ
ϑn−ϑ∗|=oP(1).
The p oo is gi en in he appendix.
5 Fini e sample p ope ies
We conside he Hilbe space H=L2([0,1]).Fo i=1,...,n he unc ional
obse a ions Xi( ), ∈[0,1], a e gene a ed acco ding o
Xi( )=1
2
5

l=1Bi,lsin  (5−Bi,l)2π−Mi,l−E[Bi,lsin (5−Bi,l)2π−Mi,l],
whe e Bi,l∼U[0,5]and Mi,l∼U[0,2π] o l=1,...,5, i=1,...,n.Us ands
o he (con inuous) uni o m dis ibu ion. The unc ional linea model is buil as
Yi=Xi( )γ3,1
3( )d +εi,
whe e he coe icien unc ion γa,b( )=ba/(a) a−1e−b I{ >0}is he densi y o
he Gamma dis ibu ion. Fu he mo e, we assume ha each Xiis obse ed on a dense,
equidis an g id o 300 e alua ion poin s.
The pa ame e es ima o s a e he egula ized es ima o s desc ibed in Example 3.3
wi h m=3 and a da a-d i en uning pa ame e λnchosen by gene alized c oss-
alida ion as desc ibed in Yuan and Cai (2010).
We model h ee simila ypes o change poin s, such ha
ε1,...,ε
n
2∼N(0,1), εn
2+1,...,ε
n∼˜
F1,δ ( espec i ely ˜
F2,δ,˜
F3,δ),
whe e ˜
F1,δ,˜
F2,δ,˜
F3,δ ha e in common ha he mean emains ze o and he a iance
emains one. In pa icula
•˜
F1,δ is he dis ibu ion unc ion o a andom a iable ha is N(−2δ, 1)dis ibu ed
wi h p obabili y 0.5 and N(2δ,1)dis ibu ed wi h p obabili y 0.5.
•˜
F2,δ is he dis ibu ion unc ion o a andom a iable ha is N(0,(1−δ)2)dis-
ibu ed wi h p obabili y 0.5 and N(0,2−(1−δ)2)dis ibu ed wi h p obabili y
0.5.
123
Tes ing o changes in he e o dis ibu ion… Page 15 o 17 33
A.4 P oo o Lemma 4.2
Fi s no e ha
ˆ
ϑn∈a g max
∈[0,1]sup
z∈R|ˆ
Gn( ,z)|=a g max
∈[0,1]#sup
z∈R
ˆ
Gn( ,z)
n1/2$.
Fu he i holds
ˆ
Gn( ,z)
n1/2=n (n−n )
n21
n 
n 

i=1
I{ˆεi≤z}− 1
n−n 
n

i=n +1
I{ˆεi≤z}
=n (n−n )
n2#1
n 
n ∧nϑ∗

i=1
I{ˆεi≤z}+I{ >ϑ
∗}1
n 
n 

i=nϑ∗+1
I{ˆεi≤z}
−1
n−n 
n

i=n ∨nϑ∗+1
I{ˆεi≤z}−I{ <ϑ
∗}1
n−n 
nϑ∗

i=n +1
I{ˆεi≤z}$
=n (n−n )
n2#n ∧nϑ∗
n F1(z)+I{ >ϑ
∗}n −nϑ∗
n F2(z)
−n−n ∨nϑ∗
n−n F2(z)−I{ <ϑ
∗}nϑ∗−n 
n−n F1(z)$+oP(1),
since we ha e
sup
∈[0,ϑ∗]
sup
z∈R
n 
n
1
n 
n 

i=1
I{ˆεi≤z}−F1(z)
≤sup
∈[0,ϑ∗]
sup
z∈R
n 
nϑ∗
1
n 
n 

i=1
I{ˆεi≤z}− 1
nϑ∗
nϑ∗

i=1
I{ˆεi≤z}
%&' (
=1
nϑ∗1/2˜
Gnϑ∗( ,z)
(A.3)
+n 
nϑ∗
1
nϑ∗
nϑ∗

i=1
I{ˆεi≤z}−F1(z)(A.4)
=oP(1).
He e we ha e used Lemma 4.1 o he e m (A.4). Fu he ˜
Gnϑ∗is de ined as ˆ
Gn
based on he iid-sample (X1,Y1), . . . , (Xk∗
n,Yk∗
n), bu whe e he esiduals a e buil
wi h ˆα,ˆ
βbased on he whole sample. Wi h he same a gumen as in he p oo o
Theo em 2.2 i holds ha
˜
Gnϑ∗( ,z)=Gnϑ∗( ,z)+oP(1)
uni o mly in ∈[0,1],z∈R,see(A.2), and hus he e m (A.3)isoP(1).
123

33 Page 16 o 17 N. Neumeye , L. Selk
Analogously one can show ha sup ∈[ϑ∗,1]supz∈R
n−n 
n1
n−n n
i=n +1I{ˆεi≤
z}−F2(z)=oP(1).
Thus, i holds uni o mly in ∈[0,1]
ˆ
Gn( ,z)
n1/2=I{ >ϑ
∗}nϑ∗(n−n )
n2(F1(z)−F2(z))
+I{ ≤ϑ∗}n (n−nϑ∗)
n2(F1(z)−F2(z))+oP(1)
=I{ >ϑ
∗}ϑ∗(1− )+I{ ≤ϑ∗} (1−ϑ∗)(F1(z)−F2(z)) +oP(1).
The asse ion hen ollows by Theo em 2.12 in Koso ok (2008)asϑ∗is well-sepa a ed
maximum o → I{ >ϑ
∗}ϑ∗(1− )+I{ ≤ϑ∗} (1−ϑ∗).
Acknowledgemen s The au ho s a e g a e ul o he Edi o s and Gues Edi o s o he o ganiza ion o he
Special Issue “Goodness-o -Fi , Change-Poin , and Rela ed P oblems”, and o he e e ees, he Associa e
Edi o and he Gues Edi o Simos Mein anis o hei cons uc i e commen s and in e es ing ideas o expand
he opic.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence,
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