Nonparametric tests for combined location-scale and Lehmann alternatives using adaptive approach and max-type metric

Funa o, Mika; Mu akami, Hide oshi; Kössle , Wol gang; Mukhe jee, Ami a a
A icle — Published Ve sion
Nonpa ame ic es s o combined loca ion-scale and
Lehmann al e na i es using adap i e app oach and max-
ype me ic
Jou nal o he Ko ean S a is ical Socie y
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Funa o, Mika; Mu akami, Hide oshi; Kössle , Wol gang; Mukhe jee, Ami a a
(2024) : Nonpa ame ic es s o combined loca ion-scale and Lehmann al e na i es using adap i e
app oach and max- ype me ic, Jou nal o he Ko ean S a is ical Socie y, ISSN 2005-2863, Sp inge
Na u e Singapo e, Singapo e, Vol. 53, Iss. 3, pp. 666-703,
h ps://doi.o g/10.1007/s42952-024-00262-7
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/316978
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1 3
RESEARCH ARTICLE
Online ISSN 2005-2863
P in ISSN 1226-3192
Nonpa ame ic es s o combined loca ion‑scale
andLehmann al e na i es using adap i e app oach
andmax‑ ype me ic
MikaFuna o1· Hide oshiMu akami2 · Wol gangKössle 3 ·
Ami a aMukhe jee4
Recei ed: 24 Augus 2023 / Accep ed: 24 Feb ua y 2024 / Published online: 2 Ap il 2024
© The Au ho (s) 2024
Abs ac
The pape deals wi h he classical wo-sample p oblem o he combined loca ion-
scale and Lehmann al e na i es, known as he e sa ile al e na i e. Recen ly, a
combina ion o he squa e o he s anda dized Wilcoxon, he s anda dized Ansa i–
B adley and he s anda dized An i-Sa age s a is ics based on he Euclidean dis ance
has been p oposed. The An i-Sa age es is he locally mos powe ul ank es o
he igh -skewed Gumbel dis ibu ion. Fu he mo e, he Sa age es is he locally
mos powe ul linea ank es o he le -skewed Gumbel dis ibu ion. Then, a
es s a is ic combining he Wilcoxon, he Ansa i–B adley, and Sa age s a is ics is
p oposed. The limi ing dis ibu ion o he p oposed s a is ic is de i ed unde he null
and he al e na i e hypo heses. In addi ion, he asymp o ic powe o he sugges ed
s a is ic is in es iga ed. Mo eo e , an adap i e es is p oposed based on a selec ion
ule. We compa e he powe pe o mance agains a ious ixed al e na i es using
Mon e Ca lo. The p oposed es s a is ic displays ou s anding pe o mance in ce ain
si ua ions. An illus a ion o he p oposed es s a is ic is p esen ed o explain a
biomedical expe imen . Finally, we o e some concluding ema ks.
Keywo ds Adap i e es · Asymp o ic powe · Maximum es
* Hide oshi Mu akami
[email p o ec ed]
1 Depa men o Applied Ma hema ics, G adua e School o Science, Tokyo Uni e si y o Science,
Tokyo, Japan
2 Depa men o Applied Ma hema ics, Tokyo Uni e si y o Science, 1-3 Kagu azaka,
Shinjyuku-ku, Tokyo162-8601, Japan
3 Depa men o Compu e Science, Humbold Uni e si y o Be lin, Be lin, Ge many
4 P oduc ion, Ope a ions andDecision Sciences A ea, XLRI-Xa ie School o Managemen ,
Jamshedpu , India
667
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 In oduc ion
Classical wo-sample compa isons be ween wo popula ions ha e ex ensi e
applica ions in many scien i ic ields, including con olled expe imen s,
biome y, psychology and indus y. Many es s exis in he li e a u e o assessing
he di e ence in a ious pa ame e s be ween wo independen popula ions.
Fu he mo e, he da a used in con ol expe imen s canno accu a ely es ima e he
popula ion dis ibu ion because he sample size is oo small, he e migh be a ew
ou lie s, o i could be a con amina ed sample. A nonpa ame ic es is p e e able
he e, as we canno assume no mali y o any o he speci ic dis ibu ion.
Many esea che s ocus on designing a es o he di e ence in a single
pa ame e , such as he loca ion, scale, o shape pa ame e in a dis ibu ion- ee
se up. Fo example, he mos amous es s o he wo-sample loca ion p oblem
in he dis ibu ion- ee se ing a e he Wilcoxon ank-sum es (Gibbons &
Chak abo i, 2021), namely
W
, o he no mal sco es es (Gibbons & Chak abo i,
2021). Assuming ha he popula ion dis ibu ion is no mal, he asymp o ic
ela i e e iciency (ARE) o he Wilcoxon ank-sum es and he no mal sco es
es ela i e o he es a e 0.955 and 1, espec i ely, indica ing ha hey a e
no in e io o he es . In addi ion, he ARE o he Wilcoxon ank-sum es o
he es is g ea e han o equal o abou 0.864 o all con inuous dis ibu ions.
The e o e, he Wilcoxon ank-sum es is s ill widely u ilized in many
applica ions, see. e.g. Le shedi e al. (2021), Lin e al. (2021) and Dao (2022).
The pa ame ic wo-sample es o es ing he a iances o he no mal
dis ibu ion is he F es . The Ansa i–B adley, namely
AB
, and Mood es s
(Gibbons & Chak abo i, 2021) a e well-known nonpa ame ic es s o he wo-
sample scale p oblem. The ARE o he Ansa i–B adley es and he Mood es
o he F es unde he assump ion o he no mal dis ibu ion a e 0.609 and 0.76,
espec i ely. Al hough ARE o he Ansa i–B adley es is lowe han ha o he
Mood es , he Ansa i–B adley is widely used in many applica ions, see, e.g.
Lahmi i (2023) and Ome e al. (2023). The wo loca ion es s, based espec i ely
on Wilcoxon and an de Wae den’s no mal sco es o wo-sample p oblems,
assume no scale di e ence in he popula ion dis ibu ions. Likewise, he wo
scale es s, based espec i ely on Ansa i–B adley and Mood sco es o wo-
sample p oblems, assume no loca ion di e ence in he popula ion dis ibu ions.
Ac ually, he e a e o en di e ences in bo h loca ion and scale in many p ac ices.
Many esea che s ocus on designing a es o he di e ence in loca ion and
scale pa ame e s be ween wo popula ions simul aneously in a dis ibu ion-
ee se up. The mos well-known es s in his ci cums ance, he es o he
loca ion and scale simul aneously, a e he Lepage (Neuhäuse , 2012) and he
Cucconi (Neuhäuse , 2012) es s. The Lepage s a is ic combines he squa e
o he s anda dized Wilcoxon ank-sum and he s anda dized Ansa i–B adley
s a is ics. The Lepage- ype s a is ic, which is a quad a ic o m o loca ion and
scale s a is ics, has been s udied by many esea che s. Fo example, Pe i (1976)
p oposed he combina ion o he squa e o s anda dized Wilcoxon ank-sum and
he squa e o s anda dized Mood s a is ics. No e ha he s a is ic o Pe i (1976)
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
is essen ially equi alen o he s a is ic o Cucconi (Neuhäuse , 2012) unde he
con inuous dis ibu ions; see Nishino and Mu akami (2019). Since Mu akami
(2011) p oposed an app oxima ion o he dis ibu ion o he s a is ic o Pe i
(1976), we can apply he app oxima ion dis ibu ion o he Cucconi s a is ic.
Fo di e en e sions o Lepage- ype s a is ics, we e e o wo ks o Büning
and Thadewald (2000), Neuhäuse (2000), Kössle (2006), Mu akami (2007),
Mu akami (2016) and Mukhe jee and Ma ozzi (2019). Recen ly, Yamaguchi and
Mu akami (2023) discussed mul i-aspec s a is ics gene alizing he Lepage- ype
s a is ics in he p esence o ies.
Kössle and Mukhe jee (2020) ecen ly no ed ha adi ional wo-sample
simul aneous es s o loca ion and scale pa ame e s a e silen abou he shape o he
dis ibu ions. A change in he shape o he dis ibu ion in a wo-sample p oblem is
add essed using he Lehmann al e na i e (Hájek e al., 1999) and is e y common in
many applica ions, see, e.g. Razzaghi (2014), Ng e al. (2021) and Chak abo y e al.
(2023). Fo he Lehmann al e na i e, o example, he An i-Sa age es (Kössle &
Mukhe jee, 2020), namely
AS
, is widely used by many esea che s. Howe e , he
di e ence in a single pa ame e is a e in many applica ions. I is mo e gene al and
ad isable o conside ha a he same ime, a shi may occu in one o mo e o he
h ee pa ame e s, namely, loca ion, scale, and shape pa ame e s. The e o e, we mus
ocus on designing app oaches o simul aneously es ing many pa ame e s be ween
wo popula ions. No e ha Kössle and Mukhe jee (2020) only conside he squa es
o Euclidean and Mahalanobis ype dis ance be ween he Wilcoxon, Ansa i–B adley
and An i-Sa age s a is ics. The An i-Sa age es is sui able o di e ence in loca ion
o he igh -skewed da a.
Howe e , we some imes encoun e le -skewed da a in p ac ical analysis. In
his case, he Sa age es , namely
S
, is one o he p e e ed es s. In he heo e ical
backg ound, he Sa age es is he locally mos powe ul ank es o he le -skewed
Gumbel dis ibu ion. Then, we may conside simila combina ions using Wilcoxon,
Ansa i–B adley and Sa age s a is ics. Also, he null dis ibu ion may be symme ic,
bu he shape al e na i e could lead o a igh o le -skewed popula ion, and he
shi di ec ion is unknown a p io i. Then, a ques ion ega ding he choice be ween
hese a ises and is add essed in he cu en pape .
Recen ly, Yamaguchi and Mu akami (2023) p oposed he ie-adjus ed e sion
o he Euclidian and Mahalanobis dis ance-based s a is ics o some s anda dized
linea ank s a is ics, ha is he mul i-aspec es s. Howe e , in p ac ical analysis,
we mus de e mine whe he o use he Wilcoxon–Ansa i–B adley–An i-Sa age
s a is ic o he Wilcoxon–Ansa i–B adley–Sa age s a is ic be o e we ea he
hypo hesis es . To his end, we migh conside a max- ype es . The la ge o he
wo s a is ics as he es s a is ic is he i s and simple way o sol e his p oblem.
Fo example, Neuhäuse e al. (2004) and Welz e al. (2018) compa ed he alidi y
o he maximum es wi h a ious nonpa ame ic es s o he wo-sample loca ion
p oblem. Addi ionally, Neuhäuse and Ho ho n (2006) discussed ha a maximum
es is an adap i e pe mu a ion es . As ano he app oach o sol e his p oblem, we
also conside an adap i e es , which selec s he es s a is ic depending on he case
g ouping. Büning (1996) p oposed an adap i e es o he mul isample loca ion
p oblem based on selec o s sugges ed by Hogg e al. (2018,pp. 622–623). In Büning
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(2000), a selec o o skewness and ail-weigh using quan ile poin s is p oposed o
his p oblem. Büning and Thadewald (2000) p oposed he wo-sample loca ion-
scale adap i e es using a selec o in oduced by Büning (2000). Fo he wo-sample
scale p oblem, Kössle (1994) p oposed he adap i e es based on new selec o s
o skewness and ail-weigh . Neuhäuse e al. (2004) p oposed an adap i e es o
he wo-sample loca ion p oblem by using he selec o s o Hogg e al. (2018,pp.
622–623). Fo he one-sample loca ion p oblem, ecen ly, Ki ani and Mu akami
(2022) p oposed an adap i e es and new selec o s. Al hough Yamaguchi and
Mu akami (2023) conside ed he mul i-aspec es s a is ic based on some linea ank
es s a is ics, maximum ype and adap i e- ype es s a e no discussed. The e o e,
we ocus on designing one maximum- ype and ano he adap i e- ype p ocedu e o
he wo-sample es ing p oblem.
The es o he pape is o ganized as ollows. We discuss s a is ical p elimina ies,
in oduce a es s a is ic and de i e he limi ing dis ibu ion o he p oposed s a is ic
unde he null hypo hesis in Sec .2. In addi ion, we de i e he limi ing dis ibu ion
o he sugges ed s a is ic unde he al e na i e hypo hesis and in es iga e asymp o ic
powe in Sec . 3.1. Sec ion 4 in oduces a maximum es and an adap i e es
based on a selec ing ule. We p esen some nume ical esul s ia Mon e Ca lo in
Sec .5. The p oposed es s a is ics a e compa ed wi h he classical omnibus es
s a is ics Kolmogo o –Smi no (Gibbons & Chak abo i, 2021), C amé – on Mises
(Ande son, 1962) and Ande son–Da ling (Pe i , 1976) as well wi h he es s a is ic
o Boos (1986). Sec ion6 is de o ed o illus a ions o he p oposed es . We o e
some concluding ema ks in Sec .7.
2 Simul aneous s a is ic o  heloca ion‑scale‑shape pa ame e s
Le
X1=(X11,…,X1n1)
and
X2=(X21,…,X2n2)
be wo andom samples o size
n1
and
n2
om absolu ely con inuous popula ions wi h he cumula i e dis ibu ion
unc ions (cd )
F1
and
F2
, espec i ely. Conside he pooled sample o size
N=n1+n2
and le
Vi
,
i=1, …,N
be 1 i he
i h
smalles o N obse a ions is om
X1
, and o he wise 0. Then, a wo-sample linea ank s a is ic is gi en by
whe e he
ai
a e app op ia e sco es. As no ed be o e, we es he loca ion
𝜇
, scale
e𝜎
and shape
e𝛿
pa ame e s a he same ime. Then we a e in e es ed in es ing he
hypo hesis
LRT
=
N
∑
i=1
aiVi
,
(1)
H
0
∶F
2
(x)=F
1
(x)
agains
H1∶F2(x)=
[
F1
(
x−𝜇
e
𝜎
)]
e𝛿
, a leas one o 𝜇≠0, 𝜎≠0, 𝛿≠
0.

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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
Unde his se up, we p opose a new s a is ic o (1) in Sec .2.1. Fu he mo e, we
de i e he limi ing null dis ibu ion o he p oposed es s a is ic in Sec .2.2.
2.1 A es s a is ic o  he e sa ile al e na i e
Kössle and Mukhe jee (2020) ecen ly no ed ha adi ional wo-sample Lepage-
ype s a is ics a e silen abou he shape o he dis ibu ions. Howe e , in a ious
applica ions, a change in he shape o he dis ibu ion along wi h he loca ion
and scale is also widesp ead. Then, Kössle and Mukhe jee (2020) p oposed he
Euclidean- ype s a is ic o he es p oblem (1) as ollows:
whe e
Rema k ha , see, e.g. Kössle and Mukhe jee (2020), he An i-Sa age es (
AS
)
is he locally mos powe ul ank es o loca ion unde he igh -skewed Gumbel
dis ibu ion wi h cd
Howe e , he e exis s he le -skewed Gumbel dis ibu ion gi en by
Then, he Sa age es is he locally mos powe ul linea ank es o loca ion i
FL
is he unde lying cd . see, e.g. Hájek e al. (1999,pp. 105–106). The e o e, in his
pape , we conside ano he ype o i-aspec s a is ic o es p oblem (1) as ollows:
T
1=
�
W−E[W]
√V[W]�2
+
�
AB −E[AB]
√V[AB]�2
+
�
AS −E[AS]
√V[AS]�2
,
W=
N
∑
i=1
iVi,E[W]=n1(N+1)
2,V[W]=n1n2(N+1)
12 ,
AB =n1(N+1)
2−
N
∑
i=1||||
i−N+1
2||||
Vi,
E[AB]={n1(N+2)
4i Nis e en ,
n1(N+1)2
4Ni Nis odd ,
V
[AB]={n1n2(N2−4)
48(N−1)i Nis e en ,
n1n1(N+1)(N2+3)
48N2i Nis odd ,
AS =
N
∑
i=1(1−
N
∑
j=i
1
j)Vi,E[AS]=0, V[AS]= n1n2
N−1(1−HN
N)
,
HN=
N
∑
j=1
1
j.
FR(x)=exp{− exp(−x)}, R(x)=exp{− exp(−x)−x},x∈ℝ.
FL(x)=1−exp{− exp(x)}, L(x)=exp{− exp(x)+x},x∈
ℝ
.
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
whe e
In addi ion, Kössle and Mukhe jee (2020) p oposed he Mahalanobis- ype s a is ic
T3
as ollows:
whe e
See, Rema k o 3.1 o Mukhe jee e al. (2021) o he de ails o he exp ession o
𝜌AB,AS
.
Simila ly o
T2
, we conside he ano he - ype o s a is ic based on he Mahalanobis
dis ance as ollows:
whe e
(2)
T
2=
�
W−E[W]
√
V[W]
�2
+
�
AB −E[AB]
√
V[AB]
�2
+
�
S−E[S]
√
V[S]
�2
,
S
=
N
∑
i=1
(N
∑
j=N+1−i
1
j
)
Vi,E[S]=n1,V[S]= n1n2
N−1
(
1−HN
N
).
(3)
T
3=TM
1
𝚺
−1
M1
T
�
M1
,
TM1=
�
W−E[W]
√V[W]
,AB −E[AB]
√V[AB]
,AS −E[AS]
√V[AS]
�
,
𝚺M1=⎛⎜⎜⎝
10𝜌W,AS
01𝜌AB,AS
𝜌W,AS 𝜌AB,AS 1⎞⎟⎟⎠
,
𝜌W,AS =√3
2�N−1
N+1�1−HN
N�−1
2
,
𝜌
AB,AS =⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
−�3N2
N2−4�1−HN
N�−1⎛⎜⎜⎝
1
2−HN+
N
2
�
j=1
1
j⎞⎟⎟⎠
i N
is e en ,
−�3(N2−1)
N2+3�1−HN
N�−1⎡
⎢⎢⎢⎣
N+3
2(N+1)−⎧
⎪
⎨
⎪
⎩
HN−
N−1
2
�
j=1
1
j⎫
⎪
⎬
⎪
⎭
⎤
⎥⎥⎥⎦
i Nis odd ,
(4)
T
4=TM
2
𝚺
−1
M
2
T
�
M
2
,
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
No e ha
T1
,
T2
,
T3
and
T4
a e special cases o Yamaguchi and Mu akami (2023).
2.2 The limi ing null dis ibu ions o 
T2
and
T4
The es s a is ic’s dis ibu ion plays a i al ole in es ing he hypo hesis. Using he
exac pe mu a ion me hod, we can de i e he exac dis ibu ion o a es s a is ic o
small sample sizes. Howe e , de i ing he exac dis ibu ion is o en di icul when
he sample sizes a e mode a e o la ge. Then, in his sec ion, we de i e he limi ing
dis ibu ions o
T2
and
T4
unde he null hypo hesis.
Le
𝜆i,i=1, 2, 3
be he eigen alues o he asymp o ic co ela ion ma ix
𝚺S
Since he eigen alues o ma ix (5) a e equal o ha o he ma ix (17) in Kössle
and Mukhe jee (2020), we immedia ely ob ain he ollowing wo heo ems by a
simila p ocedu e o ha o Kössle and Mukhe jee (2020).
Theo em1 Assume ha
n1∕N∈(0, 1)
as
min(n1,n2)
→
∞
. The limi ing null dis i-
bu ion o
T2
is app oxima ely equi alen o
1.7299Z+0.2692
, whe e
Z
∼𝜒
2
d
is a chi
squa e andom a iable wi h
d =1.5786
deg ees o eedom.
P oo See he Appendix 1.
◻
Then, as a consequence o Theo em1, we ge he ollowing co olla y.
Co olla y 1 Assume ha
n1∕N∈(0, 1)
as
min(n1,n2)
→
∞
. The le el
𝛼
c i ical poin
o
T2
can be app oxima ed by
TM2=
�
W−E[W]
√V[W]
,AB −E[AB]
√V[AB]
,S−E[S]
√V[S]
�,
𝚺M2=⎛⎜⎜⎝
10𝜌W,S
01𝜌AB,S
𝜌W,S 𝜌AB,S 1⎞⎟⎟⎠
,
𝜌W,S =𝜌W,AS,
𝜌AB,S
=−𝜌
AB,AS
.
(5)
𝚺
S=lim
min{n1,n2}→∞
𝚺M2=lim
min{n1,n2}→∞
⎛
⎜
⎜
⎝
10𝜌W,S
01𝜌AB,S
𝜌W,S 𝜌AB,S 1
⎞
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎝
10 √3
2
01
√3
2(1−2 log 2)
√
3
2
√
3
2
(1−2 log 2)1
⎞
⎟
⎟
⎟
⎟
⎠
.
Q,1−𝛼=1.7299 Z,1−𝛼+0.2692,
673
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
whe e
Z,1−𝛼
is he
1−𝛼
quan ile o he dis ibu ion o he andom a iable Z.
By eplacing
AS
in Kössle and Mukhe jee (2020) wi h
S
, we immedia ely ob ain
Lemma 1.
Lemma 1 Assume ha
n1∕N∈(0, 1)
as
min(n1,n2)
→
∞
. The asymp o ic null join
dis ibu ion o
TM2
is a i a ia e no mal wi h mean ec o (0, 0, 0) and a iance-
co a iance ma ix gi en by (5).
Theo em2 Assume ha
n1∕N∈(0, 1)
as
min(n1,n2)
→
∞
. The limi ing null dis i-
bu ion o
T4
con e ges o a Chi-squa e dis ibu ion wi h h ee deg ees o eedom.
P oo See Appendix 2.
◻
3 The dis ibu ion o  es s a is ics unde  heal e na i e hypo hesis
The sco e gene a ing unc ions o
W
,
AB
,
AS
and
S
a e espec i ely gi en by
Rema k ha he sco e gene a ing unc ion o AB in Kössle and Mukhe jee (2020)
should ha e an opposi e sign, see Kössle (2006). Howe e , i does no a ec he
eigen alues o he co ela ion ma ix and asymp o ic null dis ibu ions. Le he
pa ame e ec o
𝚯
�
N
=(𝜇 ,𝜎 ,
𝛿)∕
√
N ,
𝜆=
n
1∕
N
∈(0, 1)
as
min(
n
1,
n
2)
→
∞
and
he pa ame e ec o
𝚯�
=lim 𝚯
�
N
=(𝜇,𝜎,𝛿
)
. Le be
Sco e ∈{W, AB, AS, S}
,
Shi ∈{Loca ion, Scale, Lehmann}
and
𝜙
W
(u)=2u−1,
𝜙
AB(u)=1−2|2u−1|,
𝜙
AS(u)=1+log(u),
𝜙S(u)=−1−log(1−u)
,
u∈(0, 1).
CSco e, Shi ( )=
d
Sco e, Shi
( )
√ISco e
,
ISco e =∫1
0
𝜙2
Sco e(u)du,
dSco e, Loca ion( )=∫1
0
𝜙�
Sco e(u) (F−1(u))du,
dSco e, Scale( )=∫1
0
𝜙�
Sco e(u) (F−1(u))F−1(u)du
,
d
Sco e, Lehmann( )=−
∫
1
0
𝜙�
Sco e(u)ulog(u)du.
680
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
ha he dis ibu ion is hea y- ailed (ligh - ailed) dis ibu ion. In his pape , we eplace
he An i-Sa age es wi h he Sa age es . The ail-weigh does no play any ole he e.
The e o e, we use only one s a is ic o simplici y ins ead o
Q1
and
Q2
. Hogg e al.
(2018,pp. 623) indica ed ha he dis ibu ion’s igh ail seems longe han he le ail
when

Q1
is la ge (2 o mo e). On he o he hand, he dis ibu ion may be skewed o
he le when

Q1
<1∕
2
. We hen used hese alues as a cu o poin . In his pape , we
deno e he combined sample
V=(X1,X2)
and p opose he selec o s a is ic based on

Q1
and Table2 as ollows:
whe e
Then, we sugges wo s a is ics
AD1
and
AD2
o he adap i e es s based on a new
selec o by
whe e
No e ha he p inciple o Hogg e  al. (2018, pp. 622–623) is based on he
independence o ank and o de s a is ics o he ull sample. Howe e , in ou
selec ion ule, in

Q1(X1)
and

Q1(X2)
we use o de s a is ics o bo h single samples
sepa a ely. The e o e he independence p ope y is no sa is ied and we ha e o
check whe he he use o he (exac o asymp o ic) c i ical alues o
T1
and
T3
a e
applicable o ha o
AD1
and
AD2
.
I
(S∗, AS∗)=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
AS∗, i 
Q1(X1)>2 and 
Q1(X2)>2
o 
Q1(X1)>2 and 
Q1(X2)<1
2and 
Q1(V)>2
o 
Q1(X1)<1
2and 
Q1(X2)>2 and 
Q1(V)>2,
S∗, i 
Q1(X1)<1
2and 
Q1(X2)<1
2
o 
Q1(X1)>2 and 
Q1(X2)<1
2and 
Q1(V)<1
2
o 
Q1(X1)<1
2and 
Q1(X2)>2 and 
Q1(V)<1
2,
andomly selec AS∗o S∗, each wi h p obabili y 1
2
, o he wise
S
∗=
�
S− E[S]
√V[S] �2
, AS∗=
�
AS − E[AS]
√V[AS] �2
.
AD
1=
�
W− E[W]
√
V[W]
�2
+
�
AB − E[AB]
√
V[AB]
�2
+I(S∗, AS∗)
,
AD
2=T∗
M2
𝚺∗−1
M2
T∗�
M2
,
T
∗
M2
=
�
W−E[W]
√V[W]
,AB −E[AB]
√V[AB]
,I(
√
S∗,
√
AS∗)
�,
𝚺
∗
M2
=
⎛⎜⎜⎝
10𝜌W,I(S,AS)
01𝜌AB,I(S,AS)
𝜌W,I(S,AS)𝜌AB,I(S,AS)1
⎞⎟⎟⎠
.

681
1 3
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
5 Nume ical esul s
5.1 Robus ness
We in es iga e he obus ness o a ious dis ibu ions o p opose he adap i e es
based on he selec o s a is ic. In his pape , we compa e he pe o mances o
T1
,
T2
,
T3
,
T4
,
T(1)
max
,
T(2)
max
,
AD1
and
AD2
. We use he exac c i ical alues lis ed in Table3
when he sample sizes a e
(n1,n2)=(10, 10),(10, 5)
. On he o he hand, when he
sample sizes a e
(n1,n2)=(100, 50),(100, 100),(200, 100),(200, 200)
, we use he
95% poin o he limi ing null dis ibu ion gi en in Theo ems 1 and 2. In addi ion,
om he esul s o Kössle and Mukhe jee (2020) and Theo ems 1 and 2 in his
pape , he limi ing null dis ibu ions o
T1
and
T3
a e same as hose o
T2
and
T4
,
espec i ely. The e o e, we use he asymp o ic c i ical alues o
T1
,
T3
,
AD1
and
AD2
. Fo
T(1)
max
and
T(2)
max
, we use he es ima ed c i ical alues lis ed in Table1.
We show he ype-I e o a es (5%) o a ious es s a is ics in Table 4.
Al hough he compe ing s a is ics
T1
,
T2
,
T3
,
T4
,
T(1)
max
and
T(2)
max
a e dis ibu ion-
ee unde he null hypo hesis, he independence be ween he selec o and
AD1
o
AD2
is no gua an eed. The e o e, we in es iga ed he ype-I e o a es o
AD1
and
AD2
o ensu e hey a e sa ely used. The simula ed esul s a e based on 1,000,000
eplica ions o Mon e-Ca lo simula ions.
Table4 indica es ha he ype-I e o s o s a is ics a e a ound he signi icance
le el as expec ed because exac c i ical alues a e used o small sample sizes. Fo
la ge sample sizes, since he ype-I e o begins o con e ge, he es ima ed c i ical
alues o some selec ed sample sizes and he asymp o ic c i ical alues a e use ul.
In p ac ice, an app oxima e pe mu a ion es wi h a andom sample is possible o
e alua ing p alue o o he sample sizes. F om he nume ical esul s, we can also
see ha he null dis ibu ion o
T1
and
T2
and he null dis ibu ion o
T3
and
T4
a e
he same. Then, we can selec
T1
o
T2
and
T3
o
T4
by ou selec ing ule.
We also es ima ed ejec ion p obabili ies when popula ions de ia e om he
assumed loca ion-scale-shape amily. Howe e , we only commen on he esul s o
sa e he space. In his pape , we ocus on he ollowing pai s o dis ibu ions,
N(0, 1)
wi h
LG
(0,
√
3
𝜋
)
,
N
(0,
3
2)
wi h
6
,
U
(0,
�
3
5
)+5−
√15
10
wi h
Be(2, 2)
and
LG
(0,
3
√2𝜋
)
wi h
6
. In hese cases, bo h popula ions’ mean, a iance and skewness a e same bu
he dis ibu ions a e di e en . The e o e, ou p oposed es may no be sui able i he
al e na i e is no in he loca ion-scale-shape amily. Bu we should also no e ha he
Kolmogo o –Smi no , he C amé – on Mises o he Ande son–Da ling es s
Table 3 The exac and asymp o ic c i ical alues (
𝛼=0.05
)
–, he asymp o ic c i ical alue is no known
(n1,n2)
T1
T2
T3
T4
T(1)
max
T(2)
max
AD1
AD2
(10,10) 8.7025 8.7025 6.8516 6.8516 9.5411 7.4987 8.7025 6.8516
(10,5) 8.1898 8.1898 6.6579 6.6579 8.8160 7.2161 8.1898 6.6579
(∞,∞)
9.1715 9.1715 7.8147 7.8147 – – 9.1715 7.8147
682
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
Table 4 Type-I e o o a ious es s a is ics o a ious densi ies
(n1,n2)
N(0, 1)
GL(0, 1)
GR(0, 1)
U(0, 1)
Exp(2)
2
C(0, 1)
𝜒2
3
(10,10)
T1
0.0501 0.0502 0.0501 0.0500 0.0501 0.0502 0.0503 0.0498
T2
0.0501 0.0502 0.0500 0.0503 0.0502 0.0499 0.0503 0.0499
T3
0.0502 0.0501 0.0498 0.0499 0.0500 0.0501 0.0503 0.0500
T4
0.0502 0.0501 0.0503 0.0500 0.0499 0.0502 0.0503 0.0496
T(1)
max
0.0503 0.0500 0.0501 0.0501 0.0502 0.0499 0.0503 0.0498
T(2)
max
0.0501 0.0500 0.0499 0.0498 0.0500 0.0504 0.0503 0.0498
AD1
0.0510 0.0525 0.0525 0.0505 0.0536 0.0512 0.0512 0.0534
AD2
0.0462 0.0466 0.0465 0.0469 0.0486 0.0469 0.0484 0.0477
(10,5)
T1
0.0505 0.0507 0.0506 0.0504 0.0505 0.0506 0.0503 0.0508
T2
0.0504 0.0509 0.0505 0.0502 0.0505 0.0505 0.0501 0.0509
T3
0.0511 0.0509 0.0504 0.0505 0.0507 0.0507 0.0505 0.0506
T4
0.0508 0.0509 0.0507 0.0503 0.0505 0.0505 0.0506 0.0508
T(1)
max
0.0501 0.0505 0.0501 0.0501 0.0501 0.0500 0.0497 0.0506
T(2)
max
0.0504 0.0505 0.0504 0.0502 0.0501 0.0503 0.0502 0.0505
AD1
0.0513 0.0524 0.0522 0.0508 0.0532 0.0517 0.0510 0.0533
AD2
0.0471 0.0473 0.0471 0.0470 0.0487 0.0476 0.0488 0.0482
(100,50)
T1
0.0482 0.0482 0.0479 0.0479 0.0476 0.0481 0.0484 0.0478
T2
0.0480 0.0480 0.0478 0.0476 0.0476 0.0480 0.0486 0.0479
T3
0.0453 0.0453 0.0452 0.0452 0.0449 0.0454 0.0451 0.0449
T4
0.0455 0.0453 0.0449 0.0453 0.0449 0.0456 0.0450 0.0450
T(1)
max
0.0500 0.0501 0.0500 0.0505 0.0500 0.0500 0.0502 0.0505
T(2)
max
0.0499 0.0497 0.0503 0.0500 0.0498 0.0499 0.0498 0.0500
AD1
0.0483 0.0497 0.0493 0.0478 0.0476 0.0483 0.0490 0.0482
AD2
0.0456 0.0450 0.0447 0.0455 0.0449 0.0454 0.0458 0.0463
(100,100)
T1
0.0482 0.0483 0.0481 0.0478 0.0483 0.0484 0.0483 0.0486
T2
0.0485 0.0483 0.0483 0.0481 0.0484 0.0485 0.0484 0.0482
T3
0.0455 0.0454 0.0452 0.0454 0.0453 0.0452 0.0456 0.0451
T4
0.0454 0.0455 0.0454 0.0452 0.0452 0.0451 0.0455 0.0449
T(1)
max
0.0501 0.0499 0.0497 0.0504 0.0496 0.0497 0.0499 0.0501
T(2)
max
0.0500 0.0497 0.0496 0.0498 0.0495 0.0494 0.0495 0.0499
AD1
0.0484 0.0497 0.0495 0.0480 0.0483 0.0486 0.0485 0.0486
AD2
0.0454 0.0453 0.0449 0.0453 0.0453 0.0468 0.0474 0.0451
(200,100)
T1
0.0486 0.0485 0.0481 0.0484 0.0486 0.0485 0.0484 0.0482
T2
0.0486 0.0485 0.0481 0.0483 0.0485 0.0482 0.0483 0.0480
T3
0.0476 0.0474 0.0469 0.0467 0.0472 0.0471 0.0472 0.0472
T4
0.0474 0.0471 0.0468 0.0470 0.0471 0.0479 0.0471 0.0474
T(1)
max
0.0500 0.0500 0.0498 0.0498 0.0504 0.0501 0.0501 0.0503
T(2)
max
0.0498 0.0500 0.0499 0.0497 0.0499 0.0500 0.0496 0.0499
AD1
0.0485 0.0497 0.0492 0.0483 0.0486 0.0483 0.0486 0.0481
AD2
0.0473 0.0466 0.0463 0.0467 0.0472 0.0471 0.0482 0.0472
683
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
Table 4 (con inued)
(n1,n2)
N(0, 1)
GL(0, 1)
GR(0, 1)
U(0, 1)
Exp(2)
2
C(0, 1)
𝜒2
3
(200,200)
T1
0.0483 0.0486 0.0485 0.0485 0.0485 0.0486 0.0485 0.0484
T2
0.0483 0.0487 0.0483 0.0483 0.0483 0.0488 0.0483 0.0483
T3
0.0478 0.0468 0.0476 0.0477 0.0475 0.0476 0.0473 0.0477
T4
0.0476 0.0472 0.0477 0.0474 0.0476 0.0476 0.0475 0.0478
T(1)
max
0.0500 0.0502 0.0500 0.0499 0.0499 0.0498 0.0499 0.0500
T(2)
max
0.0500 0.0499 0.0501 0.0501 0.0500 0.0497 0.0501 0.0497
AD1
0.0484 0.0497 0.0495 0.0484 0.0485 0.0487 0.0485 0.0484
AD2
0.0478 0.0467 0.0471 0.0475 0.0475 0.0486 0.0496 0.0477
LG(0, 1)
LA(0, 1)
Ga(5, 1)
BE(2, 2)
BE(0.5, 2)
BE(2, 0.5)
CN1(0, 1, 0, 9)
CN2(1, 4, −1, 1)
(10,10)
T1
0.0495 0.0505 0.0502 0.0501 0.0500 0.0499 0.0501 0.0499
T2
0.0498 0.0502 0.0505 0.0501 0.0502 0.0499 0.0500 0.0499
T3
0.0501 0.0501 0.0498 0.0496 0.0501 0.0500 0.0504 0.0503
T4
0.0504 0.0500 0.0495 0.0500 0.0500 0.0497 0.0501 0.0509
T(1)
max
0.0497 0.0503 0.0504 0.0501 0.0501 0.0501 0.0501 0.0499
T(2)
max
0.0505 0.0500 0.0497 0.0496 0.0498 0.0497 0.0502 0.0506
AD1
0.0506 0.0516 0.0524 0.0506 0.0531 0.0530 0.0510 0.0524
AD2
0.0463 0.0457 0.0462 0.0466 0.0491 0.0487 0.0467 0.0469
(10,5)
T1
0.0509 0.0504 0.0507 0.0507 0.0511 0.0508 0.0507 0.0509
T2
0.0508 0.0502 0.0506 0.0508 0.0511 0.0505 0.0506 0.0511
T3
0.0504 0.0507 0.0504 0.0501 0.0503 0.0506 0.0506 0.0509
T4
0.0505 0.0506 0.0504 0.0507 0.0503 0.0507 0.0507 0.0507
T(1)
max
0.0505 0.0499 0.0504 0.0504 0.0508 0.0503 0.0502 0.0506
T(2)
max
0.0504 0.0504 0.0502 0.0504 0.0499 0.0502 0.0502 0.0506
AD1
0.0520 0.0516 0.0521 0.0513 0.0540 0.0536 0.0516 0.0527
AD2
0.0466 0.0465 0.0467 0.0468 0.0488 0.0492 0.0471 0.0476
(100,50)
T1
0.0479 0.0481 0.0480 0.0486 0.0480 0.0475 0.0479 0.0478
T2
0.0479 0.0479 0.0479 0.0485 0.0480 0.0475 0.0481 0.0479
T3
0.0452 0.0449 0.0449 0.0451 0.0459 0.0448 0.0450 0.0449
T4
0.0452 0.0450 0.0451 0.0453 0.0453 0.0451 0.0449 0.0450
T(1)
max
0.0501 0.0501 0.0499 0.0504 0.0498 0.0499 0.0501 0.0501
T(2)
max
0.0497 0.0497 0.0498 0.0498 0.0498 0.0498 0.0499 0.0494
AD1
0.0479 0.0482 0.0488 0.0485 0.0480 0.0475 0.0481 0.0487
AD2
0.0450 0.0446 0.0442 0.0453 0.0459 0.0451 0.0444 0.0463
684
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
designed o gene al al e na i es a e also no e y p ac ical in hese cases and u u e
esea ches on his a e highly wa an ed.
5.2 Powe compa ison
Kössle and Mukhe jee (2020) and Mukhe jee e al. (2021) compa ed he powe s
o
T1
and
T3
wi h a ious exis ing s a is ics and hey showed he alidi y o
T1
o
T3
o a ious dis ibu ions. The e o e, we ocus on compa ing he powe o
T1
,
T2
,
T3
,
T4
,
T(1)
max
,
T(2)
max
,
AD1
,
AD2
, he s a is ic o Boos (1986), abb e ia ed by
BOOS
,
he Kolmogo o –Smi no s a is ic (Gibbons & Chak abo i, 2021) (KS), he
C amé – on Mises s a is ic (Ande son, 1962) (C M), and he Ande son–Da ling
s a is ic (Pe i , 1976) (A–D), o
(n1,n2)=(10, 10)
and (10,5) in his sec ion. No e
ha he es s a is ic o Boos (1986) is a combina ion o se e al mul isample linea
ank s a is ics. I capi alizes he no ion o he Legend e polynomials o cons uc a
mul isample s a is ic. Based on he i s h ee Legend e polynomials, he es s a is ic
o Boos (1986) is a es s a is ic o loca ion, scale and skew pa ame e s as ollows:
Table 4 (con inued)
LG(0, 1)
LA(0, 1)
Ga(5, 1)
BE(2, 2)
BE(0.5, 2)
BE(2, 0.5)
CN1(0, 1, 0, 9)
CN2(1, 4, −1, 1)
(100,100)
T1
0.0480 0.0484 0.0482 0.0476 0.0482 0.0485 0.0485 0.0484
T2
0.0481 0.0483 0.0482 0.0478 0.0480 0.0484 0.0485 0.0484
T3
0.0455 0.0449 0.0451 0.0451 0.0455 0.0452 0.0456 0.0453
T4
0.0455 0.0454 0.0454 0.0448 0.0452 0.0453 0.0453 0.0456
T(1)
max
0.0498 0.0498 0.0500 0.0496 0.0499 0.0497 0.0498 0.0500
T(2)
max
0.0496 0.0500 0.0499 0.0491 0.0494 0.0492 0.0495 0.0497
AD1
0.0479 0.0484 0.0490 0.0476 0.0482 0.0484 0.0485 0.0490
AD2
0.0457 0.0455 0.0448 0.0450 0.0455 0.0453 0.0464 0.0477
(200,100)
T1
0.0479 0.0480 0.0488 0.0485 0.0479 0.0488 0.0484 0.0480
T2
0.0478 0.0482 0.0483 0.0489 0.0481 0.0486 0.0484 0.0481
T3
0.0470 0.0469 0.0471 0.0472 0.0474 0.0470 0.0476 0.0472
T4
0.0470 0.0472 0.0472 0.0470 0.0474 0.0470 0.0474 0.0470
T(1)
max
0.0498 0.0497 0.0501 0.0498 0.0502 0.0500 0.0501 0.0496
T(2)
max
0.0498 0.0494 0.0497 0.0496 0.0500 0.0497 0.0504 0.0494
AD1
0.0478 0.0481 0.0493 0.0487 0.0479 0.0486 0.0484 0.0483
AD2
0.0470 0.0472 0.0459 0.0471 0.0474 0.0470 0.0470 0.0490
(200,200)
T1
0.0484 0.0485 0.0485 0.0483 0.0482 0.0486 0.0482 0.0481
T2
0.0483 0.0483 0.0483 0.0483 0.0480 0.0482 0.0483 0.0484
T3
0.0477 0.0474 0.0473 0.0473 0.0473 0.0475 0.0478 0.0476
T4
0.0478 0.0474 0.0474 0.0475 0.0477 0.0477 0.0477 0.0475
T(1)
max
0.0502 0.0494 0.0502 0.0501 0.0500 0.0499 0.0501 0.0499
T(2)
max
0.0499 0.0494 0.0502 0.0500 0.0505 0.0497 0.0501 0.0497
AD1
0.0483 0.0484 0.0489 0.0484 0.0482 0.0482 0.0481 0.0482
AD2
0.0479 0.0475 0.0463 0.0472 0.0473 0.0477 0.0479 0.0504
685
1 3
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
The e a e b oadly se en possible ypes o shi s, h ee o which a e an isola ed shi
in one o he h ee pa ame e s, loca ion, scale and shape; ano he h ee a e mixed
shi s in ol ing any wo ou o h ee pa ame e s, and a si ua ion wi h a shi in all he
h ee pa ame e s. We use he no mal and he logis ic dis ibu ions as examples o
symme ic dis ibu ions. In addi ion, we use le -skewed and igh -skewed Gumbel
dis ibu ions as examples o asymme ic dis ibu ions. The powe pa e ns o i e
s a is ics unde a ious al e na i es a e simila o hese ou dis ibu ions. The e o e,
o sa e space, we only display he esul s o no mal, logis ic, le -skewed Gumbel
and igh -skewed Gumbel dis ibu ion. We show he esul s o he symme ic
dis ibu ions and he asymme ic dis ibu ions in Tables5, 6, 7, 8, espec i ely.
The s a is ics
T1
and
T2
(o
T3
and
T4
) di e only by he componen s AS and
S designed o de ia ions in shape pa ame e s. The e o e, he e is no di e ence in
he powe s o
T1
,
T2
,
AD1
and
T(1)
max
(o
T3
,
T4
,
AD2
and
T(2)
max
) o he pu e loca ion
o he pu e scale pa ame e in symme ic dis ibu ions wi h equal sample sizes.
F om Tables5, 6, 7 and 8, al oge he
T1
is be e han
T2
which is in acco dance
wi h he heo y since es
AS
is op imal o Lehmann al e na i es. Fu he , in
mos cases he powe o
AD1
(
AD2
) is be ween ha o
T1
and
T2
(
T3
and
T4
) as
expec ed. Tes s based on he Euclidean dis ance seem o be be e han ha based
on he Mahalanobis dis ance. The winne o ou s udy is es
T(1)
max
densely ollowed
by
T1
. The adap i e es
AD1
is on he hi d place. The o he es s a e wo se. The
max- es includes he be e o he wo es s
AS
and
S
o he cu en si ua ion. The
Adap i e es migh be conside ed as a compe i o which migh be imp o ed by a
possibly be e selec ing ule. Ne e heless, we sugges o apply he Max es
T(1)
max
as i is simple and be e han he adap i e es in many si ua ions. Fu he mo e,
compa ed o he goodness-o - i es s, in many cases, he powe o
T(1)
max
is g ea e
han o simila o he maximal powe o he h ee es s
C M
,
KS
, and A–D. In
ac , we compu ed he anks o he s a is ics acco ding o he simula ed powe . Fo
each al e na i e con igu a ion, he wo s powe is assigned he ank one, ha wi h
he la ges powe is assigned he ank 12. Thus, he es s a is ics wi h he la ges
ank sums a e he bes . In Table9, we lis he ank sums o s a is ics o e all 17
conside ed al e na i es o
(n1,n2)=(10, 10)
and (10, 5).
Table9 shows ha
T(1)
max
has he la ges ank sum o no mal, logis ic, le -skewed
Gumbel and igh -skewed Gumbel dis ibu ions. Fu he , he ank sum o
AD1
(
AD2
) is be ween ha o
T1
and
T2
(
T3
and
T4
).
BOOS
=12
n1n2(N+1)
{
N
∑
i=1
(
i−N+1
2
)
Vi
}2
+180
n1n2(N+1)(N2−4){N
∑
i=1[(i−N+1
2)2
−N2−1
12 ]Vi}
2
+7
n1n2(N+1)(N2−4)(N2−9){N
∑
i=1[20(i−N+1
2)3
−(3N2−7)
(
i−N+1
2)]
Vi
}
2
.

686
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
Table 5 Powe o a ious es s o no mal and logis ic dis ibu ions
(n1,n2)=(10, 10)
wi h
𝛼=0.05
(𝜇,𝜎,𝛿)
T1
T2
T3
T4
AD1
AD2
T(1)
max
T(2)
max
BOOS
C M
KS
A–D
The null dis ibu ion is
N(0, 1)
(1,0,0) 0.4900 0.4899 0.3418 0.3417 0.4911 0.3415 0.4848 0.3104 0.3220 0.5035 0.4087 0.5266
(0,2,0) 0.8014 0.8014 0.8804 0.8803 0.8022 0.8804 0.8299 0.8719 0.9179 0.2297 0.2903 0.4431
(0, 0, −1.5)
0.7843 0.6795 0.5705 0.6337 0.7316 0.6025 0.7453 0.5652 0.5732 0.7324 0.6841 0.7557
(1, 0, −2)
0.6535 0.4754 0.4407 0.5363 0.5639 0.4893 0.6004 0.4645 0.4567 0.5400 0.5345 0.5669
(1,0,0.5) 0.8590 0.8286 0.6944 0.7151 0.8442 0.7046 0.8413 0.6602 0.6894 0.8518 0.7751 0.8681
(1, −1, 0)
0.9131 0.7604 0.7830 0.8570 0.8360 0.8204 0.8868 0.8099 0.8024 0.8171 0.8407 0.8374
(1,1,0) 0.3117 0.5010 0.5067 0.4151 0.4068 0.4613 0.4529 0.4451 0.4790 0.2519 0.1732 0.2862
(0, 1, −1)
0.8756 0.6741 0.7642 0.8467 0.7726 0.8066 0.8426 0.7996 0.8001 0.6781 0.7495 0.7174
(0,1,1) 0.7018 0.8447 0.7376 0.6574 0.7730 0.6979 0.8076 0.6740 0.6706 0.7693 0.6359 0.7900
(1, 1, −1.5)
0.9296 0.7682 0.8485 0.9098 0.8467 0.8803 0.9071 0.8760 0.8750 0.7594 0.8327 0.7985
(1,1,1.5) 0.9996 0.9999 0.9988 0.9984 0.9997 0.9986 0.9998 0.9977 0.9984 0.9998 0.9983 0.9999
(1, −1, −1.5)
0.2196 0.1358 0.1456 0.1981 0.1789 0.1716 0.1896 0.1649 0.1729 0.1531 0.1527 0.1646
(1, −1, 1.5)
0.9970 0.9752 0.9825 0.9911 0.9859 0.9868 0.9954 0.9856 0.9820 0.9825 0.9915 0.9864
(−1, 1, −1.5)
0.9939 0.9590 0.9708 0.9848 0.9758 0.9781 0.9910 0.9759 0.9718 0.9709 0.9836 0.9768
(−1, 1, 1.5)
0.7147 0.8321 0.7038 0.6344 0.7734 0.6695 0.7961 0.6384 0.6420 0.7750 0.6403 0.7956
(−1, −1, −1.5)
0.9687 0.9886 0.9553 0.9401 0.9787 0.9477 0.9834 0.9334 0.9389 0.9798 0.9381 0.9839
(−1, −1, 1.5)
0.6000 0.8214 0.8187 0.7223 0.7097 0.7712 0.7814 0.7684 0.7874 0.5424 0.4256 0.6004
The null dis ibu ion is
LG(0, 1)
(1,0,0) 0.2042 0.2038 0.1487 0.1489 0.2057 0.1482 0.2044 0.1415 0.1394 0.2148 0.1753 0.2260
(0,2,0) 0.7712 0.7708 0.8569 0.8569 0.7707 0.8571 0.8014 0.8460 0.8966 0.2200 0.2718 0.4062
(0, 0, −1.5)
0.7843 0.6795 0.5704 0.6338 0.7275 0.6056 0.7454 0.5651 0.5730 0.7322 0.6839 0.7557
(1, 0, −2)
0.8389 0.7006 0.6452 0.7192 0.7599 0.6886 0.8008 0.6530 0.6413 0.7539 0.7470 0.7773
(1,0,0.5) 0.5833 0.5238 0.3957 0.4325 0.5556 0.4137 0.5561 0.3832 0.3949 0.5716 0.5005 0.5923
687
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
Table 5 (con inued)
(𝜇,𝜎,𝛿)
T1
T2
T3
T4
AD1
AD2
T(1)
max
T(2)
max
BOOS
C M
KS
A–D
(1, −1, 0)
0.6584 0.4303 0.5062 0.6201 0.5451 0.5636 0.6042 0.5535 0.5638 0.4604 0.5061 0.4910
(1,1,0) 0.2494 0.3686 0.4028 0.3426 0.3099 0.3731 0.3380 0.3552 0.3979 0.1538 0.1119 0.1802
(0, 1, −1)
0.8549 0.6463 0.7306 0.8189 0.7426 0.7788 0.8179 0.7664 0.7667 0.6636 0.7272 0.6988
(0,1,1) 0.6808 0.8143 0.6969 0.6209 0.7458 0.6602 0.7765 0.6327 0.6338 0.7501 0.6206 0.7692
(1, 1, −1.5)
0.9437 0.8020 0.8629 0.9177 0.8639 0.8944 0.9243 0.8855 0.8804 0.8157 0.8708 0.8433
(1,1,1.5) 0.9961 0.9983 0.9900 0.9876 0.9972 0.9889 0.9976 0.9834 0.9879 0.9980 0.9901 0.9984
(1, −1, −1.5)
0.0689 0.0850 0.0799 0.0657 0.0787 0.0723 0.0760 0.0724 0.0801 0.0634 0.0451 0.0723
(1, −1, 1.5)
0.9671 0.8647 0.9075 0.9442 0.9155 0.9262 0.9551 0.9202 0.9099 0.8741 0.9228 0.8946
(−1, 1, −1.5)
0.9865 0.9276 0.9459 0.9706 0.9532 0.9601 0.9802 0.9553 0.9503 0.9483 0.9657 0.9575
(−1, 1, 1.5)
0.8493 0.9131 0.8093 0.7690 0.8800 0.7900 0.8919 0.7574 0.7712 0.8887 0.7903 0.9011
(−1, −1, −1.5)
0.8463 0.9090 0.7938 0.7518 0.8798 0.7720 0.8853 0.7361 0.7433 0.8737 0.7550 0.8908
(−1, −1, 1.5)
0.3814 0.4831 0.5642 0.4979 0.4324 0.5316 0.4667 0.5167 0.5842 0.1717 0.1427 0.2229
688
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1 3
Table 6 Powe o a ious es s o no mal and logis ic dis ibu ions
(n1,n2)=(10, 5)
wi h
𝛼=0.05
(𝜇,𝜎,𝛿)
T1
T2
T3
T4
AD1
AD2
T(1)
max
T(2)
max
BOOS
C M
KS
A–D
The null dis ibu ion is
N(0, 1)
(1,0,0) 0.3044 0.3755 0.2035 0.2333 0.3413 0.2176 0.3403 0.1931 0.2068 0.3489 0.2838 0.3690
(0,2,0) 0.6619 0.6618 0.7244 0.7248 0.6625 0.7246 0.6985 0.7188 0.7370 0.2566 0.2364 0.2808
(0, 0, −1.5)
0.6475 0.5285 0.4392 0.4620 0.5879 0.4505 0.6075 0.3984 0.4185 0.5435 0.4883 0.5785
(1, 0, −2)
0.5375 0.4036 0.3649 0.4037 0.4704 0.3844 0.4978 0.3510 0.3628 0.3979 0.3662 0.4310
(1,0,0.5) 0.6075 0.6596 0.4212 0.4687 0.6341 0.4439 0.6275 0.3924 0.4394 0.6690 0.5836 0.6874
(1, −1, 0)
0.4739 0.4114 0.2699 0.4951 0.4437 0.3790 0.4189 0.4475 0.5059 0.6239 0.6735 0.5957
(1,1,0) 0.3216 0.4232 0.3996 0.3605 0.3733 0.3793 0.4057 0.3631 0.3806 0.2229 0.1713 0.2493
(0, 1, −1)
0.7332 0.5977 0.6280 0.6795 0.6648 0.6542 0.7046 0.6387 0.6430 0.5173 0.5039 0.5483
(0,1,1) 0.5825 0.7101 0.5642 0.5264 0.6456 0.5454 0.6741 0.5010 0.5141 0.5820 0.4761 0.6160
(1, 1, −1.5)
0.7992 0.6754 0.7077 0.7542 0.7362 0.7313 0.7740 0.7204 0.7217 0.5887 0.5768 0.6170
(1,1,1.5) 0.9869 0.9948 0.9719 0.9687 0.9908 0.9702 0.9927 0.9500 0.9611 0.9880 0.9680 0.9914
(1, −1, −1.5)
0.0699 0.0724 0.0594 0.0990 0.0720 0.0787 0.0623 0.0842 0.0790 0.0939 0.1019 0.0900
(1, −1, 1.5)
0.8060 0.7370 0.5511 0.8395 0.7715 0.6906 0.7599 0.8099 0.8609 0.9222 0.9594 0.9067
(−1, 1, −1.5)
0.9405 0.8775 0.8776 0.9007 0.9082 0.8896 0.9274 0.8780 0.8772 0.8587 0.8465 0.8747
(−1, 1, 1.5)
0.5746 0.6960 0.5258 0.4970 0.6351 0.5114 0.6583 0.4605 0.4785 0.5838 0.4742 0.6184
(−1, −1, −1.5)
0.8405 0.8480 0.7343 0.6498 0.8446 0.6902 0.8323 0.6569 0.7312 0.9083 0.8239 0.9080
(−1, −1, 1.5)
0.1729 0.2608 0.3969 0.2181 0.2178 0.3056 0.1742 0.3800 0.4252 0.3211 0.2483 0.2846
The null dis ibu ion is
LG(0, 1)
(1,0,0) 0.1294 0.1697 0.0942 0.1201 0.1508 0.1062 0.1494 0.1027 0.1045 0.1555 0.1301 0.1641
(0,2,0) 0.6339 0.6344 0.6995 0.7000 0.6349 0.6997 0.6717 0.6916 0.7124 0.2480 0.2276 0.2721
(0, 0, −1.5)
0.6467 0.5279 0.4388 0.4608 0.5840 0.4504 0.6070 0.3977 0.4175 0.5428 0.4873 0.5778
(1, 0, −2)
0.6969 0.5675 0.5110 0.5481 0.6270 0.5311 0.6594 0.4867 0.4993 0.5680 0.5270 0.6022
(1,0,0.5) 0.3291 0.3755 0.2007 0.2614 0.3537 0.2291 0.3459 0.2146 0.2320 0.3948 0.3438 0.4058
689
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
Table 6 (con inued)
(𝜇,𝜎,𝛿)
T1
T2
T3
T4
AD1
AD2
T(1)
max
T(2)
max
BOOS
C M
KS
A–D
(1, −1, 0)
0.2007 0.1596 0.1434 0.2714 0.1814 0.2054 0.1545 0.2467 0.2701 0.2747 0.3353 0.2510
(1,1,0) 0.2649 0.3282 0.3276 0.3026 0.2982 0.3149 0.3241 0.3006 0.3220 0.1591 0.1237 0.1801
(0, 1, −1)
0.7136 0.5752 0.5999 0.6525 0.6410 0.6276 0.6836 0.6076 0.6139 0.5042 0.4939 0.5344
(0,1,1) 0.5574 0.6804 0.5247 0.4939 0.6181 0.5094 0.6433 0.4651 0.4803 0.5628 0.4572 0.5949
(1, 1, −1.5)
0.8223 0.7020 0.7237 0.7687 0.7573 0.7482 0.7973 0.7320 0.7336 0.6400 0.6292 0.6666
(1,1,1.5) 0.9578 0.9781 0.9155 0.9116 0.9676 0.9134 0.9716 0.8732 0.8977 0.9626 0.9187 0.9703
(1, −1, −1.5)
0.0593 0.0432 0.0589 0.0492 0.0518 0.0537 0.0478 0.0548 0.0507 0.0511 0.0424 0.0545
(1, −1, 1.5)
0.5323 0.4390 0.3157 0.6161 0.4868 0.4585 0.4566 0.5871 0.6586 0.6989 0.8029 0.6650
(−1, 1, −1.5)
0.9148 0.8345 0.8340 0.8642 0.8713 0.8505 0.8981 0.8330 0.8332 0.8099 0.7977 0.8295
(−1, 1, 1.5)
0.6976 0.7945 0.6218 0.6047 0.7448 0.6130 0.7634 0.5551 0.5796 0.7119 0.6029 0.7415
(−1, −1, −1.5)
0.6358 0.6142 0.4745 0.3938 0.6255 0.4308 0.6069 0.3951 0.4585 0.7029 0.5755 0.7082
(−1, −1, 1.5)
0.0988 0.1248 0.2374 0.1997 0.1128 0.2186 0.0700 0.2216 0.2022 0.0734 0.0649 0.0601
696
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1 3
Table 10 AST da a
Hepa i is
33.1 67.0 164.2 187.7 37.8 39.0 45.0 96.2 60.9 31.6 48.4 32.0
53.5 77.6 31.1 39.0 38.1 132.8 324.0 63.2 16.7 38.3 46.0 114.4
Ci hosis
60.0 35.6 60.2 263.1 101.9 113.0 19.2 102.0 185.0 66.6 319.8 123.0
80.3 181.8 110.1 65.2 95.4 143.2 54.0 90.4 55.7 36.3 30.4 150.0
285.8 110.3 44.4 99.0 62.0 80.0
Tes S a is ic W AB AS S L
T1
T2
T3
T4
p alues 0.0277 0.4120 0.0524 0.1789 0.0622 0.0460 0.0850 0.0770 0.0750
Tes S a is ic
AD1
AD2
T(1)
max
T(2)
max
BOOS
C M
KS
A–D
p alues 0.0465 0.0770 0.0644 0.1052 0.0546 0.0197 0.0141 0.0279

697
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Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
Table 11 GGT da a
Hepa i is
18.9 65.0 90.4 40.2 35.9 37.0 43.0 48.1 33.1 34.4 68.2 40.6
57.9 143.4 27.6 158.2 92.1 76.4 392.2 491.0 11.5 24.7 22.3 169.8
Ci hosis
99.0 133.4 151.0 61.0 65.6 138.0 105.6 201.0 399.5 28.5 93.7 35.9
17.6 273.7 56.9 28.5 53.6 400.3 107.0 46.8 146.3 112.0 142.5 49.7
101.1 650.9 35.9 64.2 50.0 34.0
Tes S a is ic W AB AS S L
T1
T2
T3
T4
p alues 0.0703 0.5864 0.0597 0.1798 0.1675 0.0928 0.1615 0.3114 0.3115
Tes S a is ic
AD1
AD2
T(1)
max
T(2)
max
BOOS
C M
KS
A–D
p alues 0.0932 0.3114 0.1236 0.4116 0.2803 0.0769 0.0972 0.1101
698
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
and scale o GGT o hepa i is di e om hose o ci hosis. Howe e , he skewness o
GGT o hepa i is is simila o ha o ci hosis. In ac , he skewness o GGT o hepa i-
is and ci hosis a e 2.304 and 2.223, espec i ely. This case looks simila o he case o
(1, −1, 0)
o
GR(0, 1)
in Table8.
No e ha he p esen a ion o p alues is only o illus a ion o he conside ed es s.
A clea decision based on all nine es s canno be d awn. Since we sugges ed o apply
es
T(1)
max
he null hypo hesis is ejec ed o he AST da a a he 0.1 le el, bu o he
GGT da a no .
7 Concluding ema ks
Fo he pas 50 yea s, he e ha e been s udies on simul aneously es ing he wo-sample
loca ion and he scale pa ame e s. Howe e , only a ew pieces o li e a u e conside
simul aneous loca ion, scale, and shape pa ame e es ing. This pape in oduced he
dis ibu ion- ee and obus adap i e es s
AD1
and
AD2
o es o he loca ion, scale,
o shape simul aneously. The es p oposed by Kössle and Mukhe jee (2020) is a
combina ion o he s anda dized Wilcoxon, he s anda dized Ansa i–B adley and he
s anda dized An i-Sa age es s.
In his pape , we addi ionally conside es s a is ics whe e he An i-Sa age es is
eplaced wi h he Sa age es . We de i e he asymp o ic dis ibu ion o he p oposed
es s. In addi ion, we sugges a selec ion ule o an adap i e es . Mo eo e , we
conside ed max- ype es s. We in es iga e he beha io o he powe o he es s o
small sample sizes ia Mon e Ca lo simula ion. In a e age, he max- ype es based on
he Euclidean dis ances is shown o be he bes . We also discussed speci ic da a om
biomedical expe imen s.
Fo u u e esea ch, we may hink o combining he Wilcoxon ank-sum es , he
Ansa i–B adley es , he An i-Sa age es and he Sa age es as he quad-aspec es
s a is ics, as sugges ed by a e e ee. Ano he esea ch opic may be, o example, a
simple new s udy whe e he Ansa i–B adley es is eplaced wi h ano he scale es .
Mo e gene ally, o he ank sco es o loca ion and scales a e also wo h conside ing.
Appendix1: P oo o Theo em1
The s a is ic
T2
can be w i en as a quad a ic o m o h ee independen ly and iden ically
dis ibu ed s anda d no mal a iables, say
Wi
,
i=1, 2, 3
. Thus
T2
=
∑3
i=1
𝜆
i
W2
i
∶=
Q
(say). Le
Q,1−𝛼
and
Z,1−𝛼
be he
1−𝛼
quan iles;
𝜇Q
and
𝜇Z
be he means; and
𝜎Q
and
𝜎Z
be he s anda d de ia ions o he dis ibu ion o Q and Z, espec i ely. We apply
he
𝜒2
app oxima ion p oposed by Liu e al. (2009) c . also Yamaguchi and Mu akami
(2023). The eigen alues o
𝚺S
a e
𝜆1=1.9284, 𝜆2=1, 𝜆3=0.0716.
699
1 3
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
The quad a ic o m Q is app oxima ed by a
𝜒2
dis ibu ion sui ably shi ed and
scaled. The deg ees o eedom and he loca ion and scale pa ame e s a e o be
de e mined. Deno e
In ou case, we ha e
Following Liu e al. (2009), he d o Q a e gi en by
and noncen ally pa ame e by ze o. To de e mine loca ion and scale, we no e ha
Le
be expec a ion and a iance o Z, whe e
Z
∼𝜒
2
d
and
d =1.5786
. Fo app oxima ion
o he quad a ic o m Q, we shall ha e
Then, he p oo is comple ed.
◻
Appendix2: P oo o Theo em2
This asse ion ollows immedia ely om Lemma 1 and om he decomposi ion
c
k=
3
∑
i=1
𝜆k
i,k=1, 2, 3, 4, s2
1=
c
2
3
c3
2
,s2=
c4
c2
2
.
s2
1
=0.6335, and s
2
=0.6645, ha is, s
2
1
<s
2.
d
=
1
s2
1
=
1.5786,
𝜇Q
=E[Q]=c
1
=3, 𝜎
Q
=
√
V[Q]=
√
2c
2
=
3.0737.
𝜇Z=E[Z]=d =1.5786, 𝜎Z=√V[Z]=√2d =1.7769
Q−𝜇
Q
𝜎Q
≈Z−𝜇Z
𝜎Z
Q≈
𝜎Q
𝜎Z
Z−
𝜎Q
𝜎Z
𝜇Z+𝜇Q
=√2c2
√2d
Z−√2c2
√2d
⋅d +c
1
=√c2s1Z−√c2
s1
+c1
=c3
c2
Z−
c2
2
c3
+c1
≈1.7299Z+0.2692.
700
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
whe e
Y∼N(0,I)
.
◻
Appendix3: P oo o Theo em3
Recall ha
T
2=TM
2
T
�
M2
. Then, we ob ain he ollowing lemma by eplacing
𝜇j
,
j=1, 2, 3
in Lemma 4.2 o Kössle and Mukhe jee (2020) wi h
𝜸
gi en in Lemma
2.
Lemma 2 Unde he sequence
𝚯
�
N
=

𝚯�∕
√
N=(𝜇 ,𝜎 ,
𝛿)∕
√N
and wi h
n1∕
N→𝜆
∈(0, 1)
as
min(n1,n2)
→
∞
he limi ing dis ibu ion o
T′
M
2
is
N(
𝜸
�,𝚺S)
,
wi h he asymp o ic expec a ion
𝜸=(𝛾W,𝛾AB,𝛾S)=(𝛾W( ),𝛾AB( ),𝛾S( ))
wi h
Ob iously, he asymp o ic expec a ion can be w i en as
Recall
𝜆i
a e he eigen alues o
𝚺S
in (5). Le
𝚲=diag(𝜆1,𝜆2,𝜆3)
, and
U
be he
ma ix o eigen ec o s o
𝚺S
. Wi h ans o ma ion
we con e he ec o
T′
M2
o
W∼N(
𝚲−
1
2U�
𝜸
�
,
I
)
. The e o e, we ha e o he
expec a ion
Le us in oduce
T
4=TM2
𝚺−1
M
2
T�
M
2
=TM2
𝚺
−1∕2
M
2
𝚺
−1∕2
M
2
T�
M
2
=Y�Y
,
𝛾W=𝛾W( )=−
√
𝜆(1−𝜆)⋅
𝜇 dW,Loca ion +𝜎 dW, Scale +

𝛿dW, Lehmann
√IW
,
𝛾
AB =𝛾AB( )=−
√𝜆(1−𝜆)⋅
𝜇 dAB, Loca ion +𝜎 dAB, Scale +
𝛿dAB, Lehmann
√IAB
,
𝛾S=𝛾S( )=−
√
𝜆(1−𝜆)⋅
𝜇 dS, Loca ion +𝜎 dS, Scale +
𝛿dS, Lehmann
√
I
S
.
𝜸
=−
√
𝜆(1−𝜆)M
S
( )
𝚯
.
W
=𝚲−
1
2U�T�
M2
,
𝚫
∶=
⎛
⎜
⎜
⎝
Δ1
Δ2
Δ3
⎞
⎟
⎟
⎠
=E[W]=𝚲−1
2U�𝜸�=−
√𝜆(1−𝜆)𝚲−1
2U�MS( )
𝚯
=−
√
𝜆(1−𝜆)
⎛
⎜
⎜
⎝
0.4750 −0.1835 0.5092
0.3603 0.9328 0
−2.4650 0.9522 2.6425
⎞
⎟
⎟
⎠
MS( )
⎛
⎜
⎜
⎝
𝜇
𝜎

𝛿
⎞
⎟
⎟
⎠
.
701
1 3
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
In addi ion, de ine ha
i
s2
1
(𝚫)
≤
s
2
(𝚫
)
,
i
s2
1
(𝚫)>s
2
(𝚫
)
,
◻
Appendix4: P oo o Theo em4
Recall ha
T
M2=
�
W−E[W]
√
V[W],AB−E[AB]
√
V[AB],S−E[S]
√
V[S]
�
is he ec o o he h ee componen s
o
T4
,
𝜼=E[TM2]
is i s expec a ion ec o , and
𝚺M2
he co ela ion ma ix o his
ec o . Recall ha
This s a is ic is, unde he al e na i e
𝚯
, asymp o ically
𝜒2
dis ibu ed wi h h ee
deg ees o eedom and noncen ali y pa ame e
which can be seen om he decomposi ion
c
k(𝚫)=
3
∑
i=1
𝜆k
i+k
3
∑
i=1
𝜆k
iΔ2
i,s2
1(𝚫)=
c
2
3(𝚫)
c3
2
(𝚫)
,s2(𝚫)=
c4(𝚫)
c2
2
(𝚫)
.
d
(𝚫)=
1
s2
1(𝚫),
𝛽
0(𝚫)=c1(𝚫)−c2
2(𝚫)
c3(𝚫)
,
𝛽
1(𝚫)=c3(𝚫)
c
2(
𝚫
)
,
d �
(𝚫)=a
2
(𝚫)−2nc(𝚫),
a
(𝚫)= 1
s1(𝚫)−�s2
1(𝚫)−s2(𝚫)
,
nc
(𝚫)=s2
1(𝚫)a3(𝚫)−a2(𝚫),
𝛽
�
0(𝚫)=c1(𝚫)−{d �(𝚫)+nc(𝚫)}√c2(𝚫)
a(𝚫)
,
𝛽
�
1(𝚫)=
√
c2(𝚫)
a(
𝚫
)
.
T
4=TM
2
𝚺
−1
M2
T
�
M2
.
nc
( ) ∶= 𝜼𝚺
−1
S
𝜼
�
=𝜆(1−𝜆)𝚯
�
M
S
( )
�
𝚺
−1
S
( )M
S
( )𝚯
,

702
Jou nal o he Ko ean S a is ical Socie y (2024) 53:666–703
1 3
whe e
Y
=𝚺
−1∕2
M2
T�
M2
and
Y
∼N(𝚺
−1∕2
M2
𝜼�,I
)
.
◻
Acknowledgemen s The au ho s wish o hank he Edi o , Associa e Edi o and wo anonymous e iew-
e s o hei kind coope a ion o imp o e he a icle. We app ecia e he olun a y con ibu ions o he
e iewe s a o ding ime o since ely ead he ea ly e sion o he manusc ip .
Funding Open Access unding p o ided by Tokyo Uni e si y o Science.
Da a a ailabili y The da ase used in his pape is publicly a ailable, wi h e e ences p o ided in he ex .
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Con lic o in e es All au ho s ha e decla ed no Con lic o in e es .
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