Vol. 00, No. 00, Mon h 200x, 1–11
A Mon e Ca lo compa ison o h ee consis en boo s ap
p ocedu es
R. Pino Mej´ıas∗†, M.D. Jim´enez Game o†, A. Enguix Gonz´alez†
†Dp o. de Es ad´ıs ica e In es igaci´on Ope a i a, Uni e sidad de Se illa, Facul ad de
Ma em´a icas, C/ Ta ia s.n., 41.012 Se illa, Spain
(Recei ed 00 Mon h 200x; In inal o m 00 Mon h 200x)
Since boo s ap samples a e simple andom samples wi h eplacemen om he o iginal sample, he
in o ma ion con en o some boo s ap samples can be e y low. To a oid his ac , some au ho s
ha e p oposed se e al a ian s o he classical boo s ap. In his pape we conside wo o hem: he
sequen ial o Poisson boo s ap and he educed boo s ap. Bo h o hem, like o dina y boo s ap,
can yield second o de accu a e dis ibu ion es ima o s, ha is, he h ee boo s ap p ocedu es a e
asymp o ically equi alen . The ques ion ha na u ally a ises is which o hem should be used in a
p ac ical si ua ion, in o he wo ds, which o hem should be used o ini e sample sizes. To y o
answe his ques ion, we ha e ca ied ou a simula ion s udy. Al hough no me hod was ound o
exhibi bes pe o mance in all he conside ed si ua ions, some ecommenda ions a e gi en.
Keywo ds: Boo s ap, Poisson boo s ap, educed boo s ap, dis ibu ion es ima ion, ini e sample
pe o mance.
1 In oduc ion
The boo s ap, in oduced by E on [1], is a powe ul ool o nonpa ame ic es-
ima ion o sampling dis ibu ions and s anda d e o s. I may be desc ibed as
ollows. Le X= (X1, X2, . . . , Xn) be a andom sample om an unknown dis-
ibu ion F, and le Tn=Tn(X;F) be a s a is ic o in e es . Le Fnbe he em-
pi ical dis ibu ion unc ion o X1, X2, . . . , Xnand le X∗= (X∗
1, X∗
2, . . . , X∗
n)
be a andom sample d awn om Fn.X∗is called a boo s ap sample. The
boo s ap me hod es ima es he dis ibu ion o Tn h ough he condi ional
dis ibu ion o T∗
n=Tn(X∗;Fn), gi en X1, X2, . . . , Xn.
Boo s ap samples a e simple andom samples o size nselec ed wi h eplace-
men om he o iginal sample. Thus, all boo s ap samples a e no equally
in o ma i e, when he in o ma ion con en o X∗is measu ed h ough he
numbe o dis inc o iginal obse a ions in i , νn. As Rao e al. [2] asse , his
∗Co esponding au ho . E-mail: [email p o ec ed]
2R. Pino Mej´ıas e al.
a iabili y is nei he necessa y no desi able. To a oid i , hese au ho s ha e
in oduced a sequen ial sampling me hod ha keeps cons an he in o ma ion
con en o boo s ap samples, he sequen ial o Poisson boo s ap (o iginally,
hey named i sequen ial boo s ap; he name Poisson boo s ap was gi en
la e by Babu e al. [3]). I consis s o sampling sequen ially (wi h eplace-
men om he o iginal sample) un il mdis inc o iginal obse a ions appea .
To keep he in o ma ion cons an , his me hod in oduces ano he sou ce o
a iabili y: he size o he sequen ial boo s ap samples.
To educe he a iabili y o νn, keeping he size o boo s ap samples ixed,
Jim´enez-Game o e al. [4] ha e in oduced a modi ied boo s ap ha consis s
o only conside ing hose boo s ap samples sa is ying k1≤νn≤k2, o some
1≤k1≤k2≤n. They call i he educed boo s ap, since i only uses a
po ion o he se o all possible boo s ap samples. The me hod is an ex ension
o he one in oduced by Mu˜noz-Ga c´ıa e al. [5], ha akes k2=n. No e ha
o dina y boo s ap is a pa icula case o educed boo s ap wi h k1= 1 and
k2=n.
Ano he me hod o educe he a iabili y o νn, keeping he size o boo -
s ap samples ixed, can be ound in S ombe g [6]. In o de o educe he
b eakdown poin o he boo s ap a iance es ima o , which is 1/n ega dless
o he b eakdown poin o he es ima o due o he ac ha P(νn=n)>0,
S ombe g [6] p oposed o use a ”limi ed eplacemen boo s ap”, in which
all boo s ap samples selec ing he same elemen mo e han m∗ imes a e
excluded, ha is, i only conside s hose boo s ap samples wi h Ni≤m∗,
1≤i≤n, whe e Ni=ca d{j:X∗
j=Xi}, 1 ≤i≤n. As be o e, no e ha
o dina y boo s ap is a pa icula case o limi ed eplacemen boo s ap wi h
m∗=n.
Singh [7] has shown ha he boo s ap es ima o o he dis ibu ion o he
sample mean is consis en and ha i can be mo e accu a e han he ap-
p oxima ion gi en by he cen al limi heo em when highe -o de popula ion
momen s exis . Babu and Singh [8] ha e ex ended hese esul s o unc ions
o mul i a ia e means and hei s uden ized e sions. Babu e al. [3] ha e
shown ha hey a e also ue o he sequen ial boo s ap when m=E(νn).
Jim´enez-Game o e al. [9] ha e also p o en hem o he educed boo s ap,
o ce ain choices o k1and k2. The e o e, he esul s in Babu e al. [3] and
Jim´enez-Game o e al. [9] e eal ha he usual boo s ap high a iabili y o
νnis no necessa y. To ou knowledge he e is no simila esul o he limi ed
eplacemen boo s ap o S ombe g [6] so, om now on we will no conside
his me hod.
In summa y, he abo e ci ed esul s ell us ha we ha e h ee me hods
o consis en ly es ima e he dis ibu ion unc ion o s a is ics in an impo an
class ( he unc ions o mul i a ia e means and hei s uden ized e sions), ha
is, all o hem beha e simila ly o la ge samples. The ques ion ha na u ally
Compa ison o boo s ap p ocedu es 3
a ises is which o hem should be used in a p ac ical si ua ion, ha is, o
ini e sample sizes. In o de o y o answe his ques ion, we ha e ca ied ou
a simula ion s udy. Al hough i is no exhaus i e (i is impossible o simula e
all cases), we can conclude some gene al ecommenda ions om he ob ained
esul s.
The pape is o ganized as ollows. In Sec ion 2 we b ie ly e iew he me hods
o be compa ed. In Sec ion 3 we desc ibe he simula ions and display he
ob ained esul s. Sec ion 4 concludes gi ing some p ac ical ecommenda ions
in he ligh o he ob ained esul s.
2 The me hods
In his Sec ion we i s s udy he a iabili y educ ion p ope ies o he consid-
e ed me hods, and hen we e iew some p ope ies o he co esponding dis i-
bu ion es ima o s. He e we only e iew he second o de co ec ness p ope ies
because hey jus make he conside ed dis ibu ion es ima o s o ha e an edge
o e he app oxima ion by he limi ing no mal dis ibu ion.
2.1 Va iabili y educ ion p ope ies
As in he In oduc ion, le νndeno e he numbe o dis inc obse a ions in a
boo s ap sample. Then unde o dina y boo s ap,
E(νn) = n·1−µ1−1
n¶n¸≃0.632n,
a (νn) = nµ1−1
n¶n
+n(n−1) µ1−2
n¶n
−n2µ1−1
n¶2n
≃0.233n,
To educe he a iabili y o νn, he sequen ial boo s ap d aws obse a ions
om he o iginal sample wi h eplacemen un il he e a e m≃E(νn) dis-
inc obse a ions in he boo s ap sample. This way, he numbe o dis inc
obse a ions is no longe a andom a iable. Ne e heless, he me hod yields
boo s ap samples wi h andom size N. The expec a ion and he a iance o
Na e (see Rao e al. [2])
E(N) = nµ1
n+1
n−1+... +1
n−m+ 1¶=n+O(1),
4R. Pino Mej´ıas e al.
a (N) =
m
X
j=1
n(j−1)
(n−j+ 1)2=n(e−1) + O(1).
Nex esul gi es he expec a ion and he a iance o νn o he educed
boo s ap (see Appendix C o a p oo ).
P oposi ion 2.1 Le X∗be a boo s ap sample, νn=νn(X∗)and le k1, k2
be wo posi i e in ege s wi h 1≤k1≤k2≤n. Then,
E(νn|k1≤νn≤k2) = k2−(k2−k1)µ1−1
k2¶n−k1
and
a (νn|k1≤νn≤k2) = (k2−k1)(k2−k1−1) µ1−2
k2¶n−k1
+
+(k2−k1)µ1−1
k2¶n−k1
−(k2−k1)2µ1−1
k2¶2(n−k1)
.
(1)
F om (1) we ha e ha i k1=k1(n) and k2=k2(n) a e such ha (n−k1)/k2
con e ges o a eal limi l6= 0, as n→ ∞, hen a (νn|k1≤νn≤k2) =
O(k2−k1). Hence, as a (νn)≃0.233n, i k2−k1=o(n), he a iabili y
o νnin he educed boo s ap will be less han in he usual boo s ap. I
(n−k1)/k2→ ∞, as n→ ∞, his implies ha k2=o(n). As i is shown in
Jim´enez-Game o e al. [9], in his case he educed boo s ap is no consis en
and hus his choice o k1and k2is no ad isable. A simila si ua ion occu s
when (n−k1)/k2→0, as n→ ∞. Hence, o adequa e choices o k1and k2,
he educed boo s ap a iabili y o νnis less han ha o o dina y boo s ap.
2.2 Second o de accu acy o he dis ibu ion es ima o s
Le X1, X2, ..., Xnbe i.i.d. andom a iables ha ing common dis ibu ion unc-
ion F, mean µand a iance σ2. Le
¯
Xn=1
n
n
X
j=1
Xj, s2
n=1
n
n
X
j=1
(Xj−¯
Xn)2and ¯
X∗
n=1
n
n
X
j=1
X∗
j.
Th oughou his pape P∗will deno e he boo s ap condi ional p obabili y
law, gi en X1, X2, . . . , Xn.
Compa ison o boo s ap p ocedu es 5
I E|X1|3<∞and Fsa is ies C am´e ’s condi ion, Theo em 1.D in Singh [7]
shows ha
sup
xn1/2¯¯¯P∗nn1/2(¯
X∗
n−¯
Xn)/sn≤xo−
−Pnn1/2(¯
Xn−µ)/σ ≤xo¯¯¯=o(1).
(2)
This implies ha he o dina y boo s ap app oxima ion o P{n1/2(¯
Xn−
µ)/σ ≤x}is be e han he no mal app oxima ion, which has an e o o
o de O(n−1/2). Theo em 3.1 in Babu e al. [3] gi es he analogue o he abo e
esul o he Poisson boo s ap, wi h m=E(νn). Co olla y 1 in Jim´enez-
Game o e al. [9] s a es condi ions on k1and k2(1 ≤k1≤k2≤n) o (2)
o hold o he educed boo s ap dis ibu ion es ima o . These condi ions a e
he ollowing:
Condi ion C.1.a Φ(w2)−Φ(w1)≥α,∀n≥n0, o some n0∈Nand some
ixed cons an α > 0 and w1+w2=o(1) o w1→ −∞ and w2→+∞,
Condi ion C.2.aw1,w2→0,
whe e w1= (k1−1−np)n−1/2σ0−1,w2= (k2−np)n−1/2σ0−1,σ02=pq−q2,p=
1−e−1,q= 1 −pand Φ deno es he s anda d no mal cumula i e dis ibu ion
unc ion. Since P(νn≤k) = Φ(w) + o(1) wi h w= (k−np)/√nσ0(Johnson
and Ko z [10, p. 318]), i k1and k2a e such ha some o he abo e condi ions
hold, hen he p opo ion o boo s ap samples no o be conside ed because
hey ha e a small numbe o di e en elemen s (νn< k1) is (almos ) he same
as he p opo ion o boo s ap samples no o be conside ed because hey ha e
a big numbe o di e en elemen s (νn> k2).
The abo e esul s can be ex ended o he mul i a ia e case and o s a is-
ics which can be exp essed as smoo h unc ions o mul i a ia e means. Le
X1, X2, ..., Xnbe ni.i.d. d-dimensional andom ec o s wi h common dis i-
bu ion unc ion F, mean µand a iance ma ix Σ, whe e dis a ixed posi i e
in ege . Le ¯
Xnbe he sample mean and le Σnbe he sample a iance ma ix.
Le Hbe a eal alued Bo el measu able unc ion on Rd. I His di e en iable
a y, we deno e by h(y) he ec o o i s o de pa ial de i a i es o Ha
y. Le τ2=h(µ)0Σh(µ) and τ2
n=h(¯
Xn)0Σnh(¯
Xn), whe e he p ime deno es
anspose.
I EkX1k3<∞,Fsa is ies C am´e ’s condi ion, His h ee imes con inu-
ously di e en iable in a neighbo hood o µ,h(µ)6= 0 and τ > 0, Co olla y 2
in Babu and Singh [8] shows ha
sup
xn1/2¯¯¯P∗hn1/2©H(¯
X∗
n)−H(¯
Xn)ª/τn≤xi−
−Phn1/2©H(¯
Xn)−H(µ)ª/τ ≤xi¯¯¯=o(1).
(3)

6R. Pino Mej´ıas e al.
Theo em 4.1 in Babu e al. [3] and Theo em 4(b) in Jim´enez-Game o e al.
[9] gi e he analogues o he abo e esul o he Poisson boo s ap (wi h
m=E(νn)) and he educed boo s ap (wi h k1and k2sa is ying condi ions
C.1.a and C.2.a), espec i ely.
A class o s a is ics equen ly used as pi o s o app oxima e pi o s is he
s uden ized e sion o he abo e conside ed s a is ics. The esul in (3) is
also ue o his class o s a is ics. Le τ∗2
n=h(¯
X∗
n)0Σ∗
nh(¯
X∗
n), whe e Σ∗
n
is he sample a iance ma ix o he boo s ap sample. Fo a d-dimensional
andom ec o X0= (X(1), X(2), ..., X(d)), le Wbe he d(d+ 3)/2- ec o ob-
ained om Xby adjoining i he p oduc s X(i)X(j), 1 ≤i≤j≤d, ha is,
W0= (X(1), X(2), ..., X(d), X2
(1), X(1)X(2), ..., X(1)X(d), X2
(2), X(2)X(3), ..., X2
(d)).
I EkW1k3<∞,Fsa is ies C am´e ’s condi ion, His h ee imes con inuously
di e en iable in a neighbo hood o µ,h(µ)6= 0 and τ > 0, hen Theo em 4 in
Babu and Singh [8] shows ha
sup
xn−1/2¯¯¯P∗hn1/2©H(¯
X∗
n)−H(¯
Xn)ª/τ∗
n≤xi−
−Phn1/2©H(¯
Xn)−H(µ)ª/τn≤xi¯¯¯=o(1).
(4)
Theo em 4.3 in Babu e al. [3] and Theo em 5(b) in Jim´enez-Game o e al.
[9] gi e he analogues o he esul in (4) o he Poisson boo s ap (wi h
m=E(νn)) and he educed boo s ap (wi h k1and k2sa is ying condi ions
C.1.a and C.2.a), espec i ely.
As we said in he In oduc ion, Rao e al. [2] asse ed ha he a iabili y o
νnis no desi able. An example o his ac eme ges when one conside s s a is-
ics which a e a s uden ized e sion o a smoo h unc ion o sample means, as
he ones in ol ed in (4), since boo s ap samples wi h all hei componen s
equal yield a degene a e alue o he conside ed s a is ics. This undesi able
si ua ion may be a oided by using he sequen ial boo s ap o he educed
boo s ap wi h some k1>1.
3 Simula ions
In Sec ion 2 we ha e seen ha he o dina y boo s ap, he sequen ial boo s ap
wi h m=E(νn) and he educed boo s ap o adequa e choices o k1and k2
yield dis ibu ion es ima o s ha ing all o hem he same asymp o ic accu acy.
To s udy and compa e he ini e sample pe o mance o he co esponding es i-
ma o s we ha e ca ied ou a simula ion expe imen . The conside ed me hods
a e displayed in Table 1, whe e [x] deno es he g ea es in ege less o equal
han x. The Appendices A and B desc ibe he employed algo i hms o gen-
e a e educed boo s ap samples and Poisson boo s ap samples, espec i ely.
Compa ison o boo s ap p ocedu es 7
Table 1. Me hods.
me hod desc ip ion
1 o dina y boo s ap
2 educed boo s ap wi h k1= [np −2√npq] + 1, k2= [np + 2√npq]
3 educed boo s ap wi h k1= [np −√npq] + 1, k2= [np +√npq]
4 educed boo s ap wi h k1=k2= [np]+1
5 sequen ial boo s ap wi h m= [np]+1
To s udy he co esponding dis ibu ion es ima o s o he s a is ic
1(X) = √n¯
Xn−µ
sn
,
we ha e gene a ed M= 1000 samples o size n= 10 om a s anda d no mal
popula ion, N(0,1). Fo each me hod in Table 1 and om each sample Xm,
1≤m≤M, we ha e gene a ed B= 1000 boo s ap samples, X∗m,1, ..., X∗m,B.
We ha e conside ed he ollowing in e als
(−∞,−3),[−3,−2.9),[−2.9,−2.8),[−2.8,−2.7), ..., [2.9,3),[3,∞) (5)
and, o each me hod, we ha e calcula ed (I, m), he ac ion o 1(X∗m,b),
1≤b≤B, alling on in e al I. As a global measu e o he bias o each
dis ibu ion es ima o we ha e conside ed
BS =X
I∈I
bias2(I),
whe e Iis he se o he in e als in (5), bias(I) = 1
MPM
m=1 (I, m)− (I) and
(I) gi es he ac ion o T(Xm), 1 ≤m≤M, alling on he in e al I; and
as a global measu e o he mean squa ed e o o each dis ibu ion es ima o
we ha e conside ed
MS =X
I∈I
mse(I),
whe e mse(I) = 1
MPM
m=1 { (I, m)− (I)}2.
We ha e epea ed he abo e expe imen o n= 20,50,100,200,500,1000
and also o he s a is ics
(X) = √nH(¯
Xn)−H(µ)
|h(¯
Xn)|sn
,
wi h H(x) = exp(x) (s a is ic 2) and H(x) = x2(s a is ic 3). Fo s a is ic 3,
8R. Pino Mej´ıas e al.
Table 2. Resul s o s a is ic 1wi h da a om a N(0,1) popula ion.
sample size
me hod 10 20 50 100 200 500 1000
1BS ×1030.61 0.98 0.63 1.26 1.11 0.66 1.05
MS ×1032.07 2.00 1.61 2.24 2.09 1.64 2.01
2BS ×1030.54 0.96 0.64 1.28 1.10 0.66 1.06
MS ×1032.07 2.00 1.63 2.23 2.07 1.64 2.04
3BS ×1030.53 0.96 0.64 1.28 1.10 0.65 1.03
MS ×1032.07 2.00 1.61 2.25 2.07 1.63 2.01
4BS ×1038.44 3.69 2.58 3.34 2.85 2.30 2.59
MS ×10310.96 4.68 3.56 4.30 3.80 3.26 3.55
5BS ×1030.64 0.98 0.63 1.26 1.12 0.66 1.06
MS ×1031.94 2.00 1.60 2.23 2.10 1.65 2.03
Table 3. Resul s o s a is ic 2wi h da a om a N(0,1) popula ion.
sample size
me hod 10 20 50 100 200 500 1000
1BS ×1031.00 0.92 0.99 0.92 0.99 1.26 0.53
MS ×1032.98 2.02 1.98 1.90 1.98 2.23 1.51
2BS ×1031.17 0.85 0.99 0.92 0.97 1.24 0.54
MS ×1033.29 2.01 1.99 1.90 1.94 2.21 1.51
3BS ×1031.19 0.85 0.99 0.91 0.98 1.27 0.53
MS ×1033.34 2.00 2.00 1.88 1.95 2.23 1.51
4BS ×1039.62 6.59 3.67 3.92 3.36 2.21 2.21
MS ×10312.53 7.61 4.65 4.87 4.31 3.17 3.18
5BS ×1030.66 0.99 0.98 0.94 0.98 1.24 0.53
MS ×1032.50 2.10 1.98 1.91 1.96 2.21 1.49
Table 4. Resul s o s a is ic 3wi h da a om a N(5,1) popula ion.
sample size
me hod 10 20 50 100 200 500 1000
1BS ×1031.02 0.63 1.45 0.69 0.86 0.84 0.92
MS ×1032.52 1.66 2.42 1.65 1.83 1.82 1.89
2BS ×1031.19 0.65 1.42 0.70 0.84 0.82 0.90
MS ×1032.76 1.68 2.40 1.67 1.83 1.79 1.88
3BS ×1031.17 0.65 1.45 0.71 0.86 0.84 0.89
MS ×1032.73 1.69 2.41 1.68 1.84 1.82 1.86
4BS ×1036.39 3.40 3.70 2.30 2.79 2.18 2.15
MS ×1038.93 4.38 4.67 3.27 3.75 3.14 3.11
5BS ×1030.96 0.62 1.45 0.69 0.83 0.82 0.89
MS ×1032.27 1.65 2.42 1.66 1.79 1.80 1.87
o a oid h(µ) = 0, he da a comes om a N(5, 1) popula ion. The ob ained
esul s a e displayed in Tables 2, 3 and 4. Tables 5, 6 and 7 p esen he esul s
o a simila expe imen wi h da a om a mix u e o wo no mal popula ions,
0.2N(0,1) + 0.8N(5,22).
All compu a ions in his pape ha e been pe o med using p og ams w i en
in he R language (R De elopmen Co e Team [11]).
Compa ison o boo s ap p ocedu es 9
Table 5. Resul s o s a is ic 1wi h da a om a 0.2N(0,1) + 0.8N(5,22) popula-
ion. sample size
me hod 10 20 50 100 200 500 1000
1BS ×1031.05 0.83 0.99 1.18 0.65 1.06 1.24
MS ×1032.51 1.83 1.96 2.16 1.62 2.02 2.21
2BS ×1030.91 0.86 0.92 1.17 0.66 1.02 1.22
MS ×1032.46 1.87 1.90 2.14 1.64 1.98 2.19
3BS ×1030.93 0.86 0.93 1.16 0.67 1.03 1.24
MS ×1032.47 1.87 1.92 2.12 1.63 2.01 2.21
4BS ×1038.95 3.22 3.80 3.47 2.05 3.12 3.10
MS ×10311.71 4.21 4.78 4.43 3.02 4.09 4.05
5BS ×1031.06 0.82 0.98 1.20 0.64 1.02 1.25
MS ×1032.34 1.83 1.94 2.18 1.61 2.00 2.22
Table 6. Resul s o s a is ic 2wi h da a om a 0.2N(0,1) + 0.8N(5,22) popula-
ion. sample size
me hod 10 20 50 100 200 500 1000
1BS ×1030.67 0.92 0.72 0.99 1.00 1.11 0.74
MS ×1032.75 2.17 1.73 1.97 1.98 2.07 1.71
2BS ×1030.75 1.02 0.80 1.08 1.00 1.09 0.72
MS ×1033.04 2.31 1.82 2.06 1.95 2.05 1.69
3BS ×1030.78 0.98 0.82 1.07 1.00 1.10 0.73
MS ×1033.00 2.28 1.83 2.04 1.97 2.06 1.71
4BS ×10316.18 6.77 3.54 3.15 3.44 3.38 2.24
MS ×10319.51 7.89 4.54 4.11 4.39 4.36 3.21
5BS ×1030.95 0.90 0.72 0.98 1.02 1.09 0.73
MS ×1032.88 2.14 1.74 1.97 1.98 2.06 1.71
Table 7. Resul s o s a is ic 3wi h da a om a 0.2N(0,1) + 0.8N(5,22) popula-
ion. sample size
me hod 10 20 50 100 200 500 1000
1BS ×1031.03 0.60 0.96 0.79 1.08 1.37 0.80
MS ×1032.87 1.71 1.96 1.76 2.03 2.35 1.75
2BS ×1031.07 0.61 0.98 0.79 1.07 1.34 0.81
MS ×1033.03 1.74 1.97 1.76 2.04 2.32 1.77
3BS ×1031.07 0.62 0.97 0.78 1.07 1.33 0.81
MS ×1033.03 1.73 1.96 1.74 2.05 2.30 1.77
4BS ×10311.66 4.19 3.51 3.34 3.05 4.01 3.07
MS ×10314.65 5.19 4.48 4.30 4.01 4.97 4.03
5BS ×1031.44 0.55 0.97 0.81 1.05 1.36 0.81
MS ×1033.11 1.65 1.97 1.79 2.01 2.34 1.78
4 Conclusions
Looking a Tables 2–7 we see ha me hod 4 has he bigges bias o all he
conside ed cases. Fo he es o he me hods in Table 1, we obse e ha he e
is no me hod beha ing much be e han he o he s o all s a is ics, all sample
sizes and all dis ibu ions. As expec ed, he di e ences in BS and MS dec ease
as he sample size inc eases. Al hough he pe o mance o me hods 1, 2, 3, 5