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

Funato, Mika; Murakami, Hidetoshi; Kössler, Wolfgang; Mukherjee, Amitava Article — Published Version Nonparametric tests for combined location-scale and Lehmann alternatives using adaptive approach and maxtype metric Journal of the Korean Statistical Society Provided in Cooperation with: Springer Nature Suggested Citation: Funato, Mika; Murakami, Hidetoshi; Kössler, Wolfgang; Mukherjee, Amitava (2024) : Nonparametric tests for combined location-scale and Lehmann alternatives using adaptive approach and max-type metric, Journal of the Korean Statistical Society, ISSN 2005-2863, Springer Nature Singapore, Singapore, Vol. 53, Iss. 3, pp. 666-703, https://doi.org/10.1007/s42952-024-00262-7 This Version is available at: https://hdl.handle.net/10419/316978 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol:.(1234567890) Journal of the Korean Statistical Society (2024) 53:666–703 https://doi.org/10.1007/s42952-024-00262-7 1 3 RESEARCH ARTICLE Online ISSN 2005-2863 Print ISSN 1226-3192 Nonparametric tests forcombined location‑scale andLehmann alternatives using adaptive approach andmax‑type metric MikaFunato1· HidetoshiMurakami2 · WolfgangKössler3 · AmitavaMukherjee4 Received: 24 August 2023 / Accepted: 24 February 2024 / Published online: 2 April 2024 © The Author(s) 2024 Abstract The paper deals with the classical two-sample problem for the combined locationscale and Lehmann alternatives, known as the versatile alternative. Recently, a combination of the square of the standardized Wilcoxon, the standardized Ansari– Bradley and the standardized Anti-Savage statistics based on the Euclidean distance has been proposed. The Anti-Savage test is the locally most powerful rank test for the right-skewed Gumbel distribution. Furthermore, the Savage test is the locally most powerful linear rank test for the left-skewed Gumbel distribution. Then, a test statistic combining the Wilcoxon, the Ansari–Bradley, and Savage statistics is proposed. The limiting distribution of the proposed statistic is derived under the null and the alternative hypotheses. In addition, the asymptotic power of the suggested statistic is investigated. Moreover, an adaptive test is proposed based on a selection rule. We compare the power performance against various fixed alternatives using Monte Carlo. The proposed test statistic displays outstanding performance in certain situations. An illustration of the proposed test statistic is presented to explain a biomedical experiment. Finally, we offer some concluding remarks. Keywords Adaptive test· Asymptotic power· Maximum test * Hidetoshi Murakami [email protected] 1 Department ofApplied Mathematics, Graduate School ofScience, Tokyo University ofScience, Tokyo, Japan 2 Department ofApplied Mathematics, Tokyo University ofScience, 1-3 Kagurazaka, Shinjyuku-ku, Tokyo162-8601, Japan 3 Department ofComputer Science, Humboldt University ofBerlin, Berlin, Germany 4 Production, Operations andDecision Sciences Area, XLRI-Xavier School ofManagement, Jamshedpur, India 667 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 1 Introduction Classical two-sample comparisons between two populations have extensive applications in many scientific fields, including controlled experiments, biometry, psychology and industry. Many tests exist in the literature for assessing the difference in various parameters between two independent populations. Furthermore, the data used in control experiments cannot accurately estimate the population distribution because the sample size is too small, there might be a few outliers, or it could be a contaminated sample. A nonparametric test is preferable here, as we cannot assume normality or any other specific distribution. Many researchers focus on designing a test for the difference in a single parameter, such as the location, scale, or shape parameter in a distribution-free setup. For example, the most famous tests for the two-sample location problem in the distribution-free setting are the Wilcoxon rank-sum test (Gibbons & Chakraborti, 2021), namely W , or the normal scores test (Gibbons & Chakraborti, 2021). Assuming that the population distribution is normal, the asymptotic relative efficiency (ARE) of the Wilcoxon rank-sum test and the normal scores test relative to the t test are 0.955 and 1, respectively, indicating that they are not inferior to the t test. In addition, the ARE of the Wilcoxon rank-sum test to the t test is greater than or equal to about 0.864 for all continuous distributions. Therefore, the Wilcoxon rank-sum test is still widely utilized in many applications, see. e.g. Letshedi etal. (2021), Lin etal. (2021) and Dao (2022). The parametric two-sample test for testing the variances of the normal distribution is the F test. The Ansari–Bradley, namely AB , and Mood tests (Gibbons & Chakraborti, 2021) are well-known nonparametric tests for the twosample scale problem. The ARE of the Ansari–Bradley test and the Mood test to the F test under the assumption of the normal distribution are 0.609 and 0.76, respectively. Although ARE of the Ansari–Bradley test is lower than that of the Mood test, the Ansari–Bradley is widely used in many applications, see, e.g. Lahmiri (2023) and Omer etal. (2023). The two location tests, based respectively on Wilcoxon and van der Waerden’s normal scores for two-sample problems, assume no scale difference in the population distributions. Likewise, the two scale tests, based respectively on Ansari–Bradley and Mood scores for twosample problems, assume no location difference in the population distributions. Actually, there are often differences in both location and scale in many practices. Many researchers focus on designing a test for the difference in location and scale parameters between two populations simultaneously in a distributionfree setup. The most well-known tests in this circumstance, the test for the location and scale simultaneously, are the Lepage (Neuhäuser, 2012) and the Cucconi (Neuhäuser, 2012) tests. The Lepage statistic combines the square of the standardized Wilcoxon rank-sum and the standardized Ansari–Bradley statistics. The Lepage-type statistic, which is a quadratic form of location and scale statistics, has been studied by many researchers. For example, Pettitt (1976) proposed the combination of the square of standardized Wilcoxon rank-sum and the square of standardized Mood statistics. Note that the statistic of Pettitt (1976) 668 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 is essentially equivalent to the statistic of Cucconi (Neuhäuser, 2012) under the continuous distributions; see Nishino and Murakami (2019). Since Murakami (2011) proposed an approximation to the distribution of the statistic of Pettitt (1976), we can apply the approximation distribution to the Cucconi statistic. For different versions of Lepage-type statistics, we refer to works of Büning and Thadewald (2000), Neuhäuser (2000), Kössler (2006), Murakami (2007), Murakami (2016) and Mukherjee and Marozzi (2019). Recently, Yamaguchi and Murakami (2023) discussed multi-aspect statistics generalizing the Lepage-type statistics in the presence of ties. Kössler and Mukherjee (2020) recently noted that traditional two-sample simultaneous tests for location and scale parameters are silent about the shape of the distributions. A change in the shape of the distribution in a two-sample problem is addressed using the Lehmann alternative (Hájek etal., 1999) and is very common in many applications, see, e.g. Razzaghi (2014), Ng etal. (2021) and Chakraborty etal. (2023). For the Lehmann alternative, for example, the Anti-Savage test (Kössler & Mukherjee, 2020), namely AS , is widely used by many researchers. However, the difference in a single parameter is rare in many applications. It is more general and advisable to consider that at the same time, a shift may occur in one or more of the three parameters, namely, location, scale, and shape parameters. Therefore, we must focus on designing approaches for simultaneously testing many parameters between two populations. Note that Kössler and Mukherjee (2020) only consider the squares of Euclidean and Mahalanobis type distance between the Wilcoxon, Ansari–Bradley and Anti-Savage statistics. The Anti-Savage test is suitable for difference in location of the right-skewed data. However, we sometimes encounter left-skewed data in practical analysis. In this case, the Savage test, namely S , is one of the preferred tests. In the theoretical background, the Savage test is the locally most powerful rank test for the left-skewed Gumbel distribution. Then, we may consider similar combinations using Wilcoxon, Ansari–Bradley and Savage statistics. Also, the null distribution may be symmetric, but the shape alternative could lead to a right or left-skewed population, and the shift direction is unknown a priori. Then, a question regarding the choice between these arises and is addressed in the current paper. Recently, Yamaguchi and Murakami (2023) proposed the tie-adjusted version of the Euclidian and Mahalanobis distance-based statistics of some standardized linear rank statistics, that is the multi-aspect tests. However, in practical analysis, we must determine whether to use the Wilcoxon–Ansari–Bradley–Anti-Savage statistic or the Wilcoxon–Ansari–Bradley–Savage statistic before we treat the hypothesis test. To this end, we might consider a max-type test. The larger of the two statistics as the test statistic is the first and simple way to solve this problem. For example, Neuhäuser etal. (2004) and Welz etal. (2018) compared the validity of the maximum test with various nonparametric tests for the two-sample location problem. Additionally, Neuhäuser and Hothorn (2006) discussed that a maximum test is an adaptive permutation test. As another approach to solve this problem, we also consider an adaptive test, which selects the test statistic depending on the case grouping. Büning (1996) proposed an adaptive test for the multisample location problem based on selectors suggested by Hogg etal. (2018,pp. 622–623). In Büning 669 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 (2000), a selector of skewness and tail-weight using quantile points is proposed for this problem. Büning and Thadewald (2000) proposed the two-sample locationscale adaptive test using a selector introduced by Büning (2000). For the two-sample scale problem, Kössler (1994) proposed the adaptive test based on new selectors for skewness and tail-weight. Neuhäuser etal. (2004) proposed an adaptive test for the two-sample location problem by using the selectors of Hogg etal. (2018,pp. 622–623). For the one-sample location problem, recently, Kitani and Murakami (2022) proposed an adaptive test and new selectors. Although Yamaguchi and Murakami (2023) considered the multi-aspect test statistic based on some linear rank test statistics, maximum type and adaptive-type tests are not discussed. Therefore, we focus on designing one maximum-type and another adaptive-type procedure for the two-sample testing problem. The rest of the paper is organized as follows. We discuss statistical preliminaries, introduce a test statistic and derive the limiting distribution of the proposed statistic under the null hypothesis in Sect.2. In addition, we derive the limiting distribution of the suggested statistic under the alternative hypothesis and investigate asymptotic power in Sect. 3.1. Section 4 introduces a maximum test and an adaptive test based on a selecting rule. We present some numerical results via Monte Carlo in Sect.5. The proposed test statistics are compared with the classical omnibus test statistics Kolmogorov–Smirnov (Gibbons & Chakraborti, 2021), Cramér–von Mises (Anderson, 1962) and Anderson–Darling (Pettitt, 1976) as well with the test statistic of Boos (1986). Section6 is devoted to illustrations of the proposed test. We offer some concluding remarks in Sect.7. 2 Simultaneous statistic forthelocation‑scale‑shape parameters Let X1=(X11,…,X1n1) and X2=(X21,…,X2n2) be two random samples of size n1 and n2 from absolutely continuous populations with the cumulative distribution functions (cdf) F1 and F2 , respectively. Consider the pooled sample of size N=n1+n2 and let Vi , i=1, …,N be 1 if the ith smallest of N observations is from X1 , and otherwise 0. Then, a two-sample linear rank statistic is given by where the ai are appropriate scores. As noted before, we test the location 𝜇 , scale e𝜎 and shape e𝛿 parameters at the same time. Then we are interested in testing the hypothesis LRT = N ∑ i=1 aiVi , (1) H 0 ∶F 2 (x)=F 1 (x) against H1∶F2(x)= [ F1 ( x−𝜇 e 𝜎 )] e𝛿 , at least one of 𝜇≠0, 𝜎≠0, 𝛿≠ 0. 670 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 Under this setup, we propose a new statistic for (1) in Sect.2.1. Furthermore, we derive the limiting null distribution of the proposed test statistic in Sect.2.2. 2.1 A test statistic fortheversatile alternative Kössler and Mukherjee (2020) recently noted that traditional two-sample Lepagetype statistics are silent about the shape of the distributions. However, in various applications, a change in the shape of the distribution along with the location and scale is also widespread. Then, Kössler and Mukherjee (2020) proposed the Euclidean-type statistic for the test problem (1) as follows: where Remark that, see, e.g. Kössler and Mukherjee (2020), the Anti-Savage test ( AS ) is the locally most powerful rank test for location under the right-skewed Gumbel distribution with cdf However, there exists the left-skewed Gumbel distribution given by Then, the Savage test is the locally most powerful linear rank test for location if FL is the underlying cdf. see, e.g. Hájek etal. (1999,pp. 105–106). Therefore, in this paper, we consider another type of tri-aspect statistic for test problem (1) as follows: 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) 4if Nis even , n1(N+1)2 4Nif Nis odd , V [AB]={n1n2(N2−4) 48(N−1)if Nis even , n1n1(N+1)(N2+3) 48N2if 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)},fR(x)=exp{− exp(−x)−x},x∈ℝ. FL(x)=1−exp{− exp(x)},fL(x)=exp{− exp(x)+x},x∈ ℝ . 671 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 where In addition, Kössler and Mukherjee (2020) proposed the Mahalanobis-type statistic T3 as follows: where See, Remark of 3.1 of Mukherjee etal. (2021) for the details of the expression of 𝜌AB,AS . Similarly to T2 , we consider the another-type of statistic based on the Mahalanobis distance as follows: where (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⎞⎟⎟⎠ if N is even , −�3(N2−1) N2+3�1−HN N�−1⎡ ⎢⎢⎢⎣ N+3 2(N+1)−⎧ ⎪ ⎨ ⎪ ⎩ HN− N−1 2 � j=1 1 j⎫ ⎪ ⎬ ⎪ ⎭ ⎤ ⎥⎥⎥⎦ if Nis odd , (4) T 4=TM 2 𝚺 −1 M 2 T � M 2 , 672 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 Note that T1 , T2 , T3 and T4 are special cases of Yamaguchi and Murakami (2023). 2.2 The limiting null distributions of T2 and T4 The test statistic’s distribution plays a vital role in testing the hypothesis. Using the exact permutation method, we can derive the exact distribution of a test statistic for small sample sizes. However, deriving the exact distribution is often difficult when the sample sizes are moderate to large. Then, in this section, we derive the limiting distributions of T2 and T4 under the null hypothesis. Let 𝜆i,i=1, 2, 3 be the eigenvalues of the asymptotic correlation matrix 𝚺S Since the eigenvalues of matrix (5) are equal to that of the matrix (17) in Kössler and Mukherjee (2020), we immediately obtain the following two theorems by a similar procedure to that of Kössler and Mukherjee (2020). Theorem1 Assume that n1∕N∈(0, 1) as min(n1,n2) → ∞ . The limiting null distribution of T2 is approximately equivalent to 1.7299Z+0.2692 , where Z ∼𝜒 2 df is a chi square random variable with df =1.5786 degrees of freedom. Proof See the Appendix 1. ◻ Then, as a consequence of Theorem1, we get the following corollary. Corollary 1 Assume that n1∕N∈(0, 1) as min(n1,n2) → ∞ . The level 𝛼 critical point of T2 can be approximated 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 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . tQ,1−𝛼=1.7299tZ,1−𝛼+0.2692, 673 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 where tZ,1−𝛼 is the 1−𝛼 quantile of the distribution of the random variable Z. By replacing AS in Kössler and Mukherjee (2020) with S , we immediately obtain Lemma 1. Lemma 1 Assume that n1∕N∈(0, 1) as min(n1,n2) → ∞ . The asymptotic null joint distribution of TM2 is a trivariate normal with mean vector (0, 0, 0) and variancecovariance matrix given by (5). Theorem2 Assume that n1∕N∈(0, 1) as min(n1,n2) → ∞ . The limiting null distribution of T4 converges to a Chi-square distribution with three degrees of freedom. Proof See Appendix 2. ◻ 3 The distribution oftest statistics underthealternative hypothesis The score generating functions of W , AB , AS and S are respectively given by Remark that the score generating function of AB in Kössler and Mukherjee (2020) should have an opposite sign, see Kössler (2006). However, it does not affect the eigenvalues of the correlation matrix and asymptotic null distributions. Let the parameter vector 𝚯 � N =(𝜇 ,𝜎 , 𝛿)∕ √ N , 𝜆= n 1∕ N ∈(0, 1) as min( n 1, n 2) → ∞ and the parameter vector 𝚯� =lim 𝚯 � N =(𝜇,𝜎,𝛿 ) . Let be Score ∈{W, AB, AS, S} , Shift ∈{Location, 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). CScore, Shift(f)= d Score, Shift (f) √IScore , IScore =∫1 0 𝜙2 Score(u)du, dScore, Location(f)=∫1 0 𝜙� Score(u)f(F−1(u))du, dScore, Scale(f)=∫1 0 𝜙� Score(u)f(F−1(u))F−1(u)du , d Score, Lehmann(f)=− ∫ 1 0 𝜙� Score(u)ulog(u)du. 680 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 that the distribution is heavy-tailed (light-tailed) distribution. In this paper, we replace the Anti-Savage test with the Savage test. The tail-weight does not play any role here. Therefore, we use only one statistic for simplicity instead of Q1 and Q2 . Hogg etal. (2018,pp. 623) indicated that the distribution’s right tail seems longer than the left tail when  Q1 is large (2 or more). On the other hand, the distribution may be skewed to the left when  Q1 <1∕ 2 . We then used these values as a cutoff point. In this paper, we denote the combined sample V=(X1,X2) and propose the selector statistic based on  Q1 and Table2 as follows: where Then, we suggest two statistics AD1 and AD2 for the adaptive tests based on a new selector by where Note that the principle of Hogg et al. (2018, pp. 622–623) is based on the independence of rank and order statistics of the full sample. However, in our selection rule, in  Q1(X1) and  Q1(X2) we use order statistics of both single samples separately. Therefore the independence property is not satisfied and we have to check whether the use of the (exact or asymptotic) critical values for T1 and T3 are applicable to that for AD1 and AD2 . I (S∗, AS∗)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ AS∗, if  Q1(X1)>2 and  Q1(X2)>2 or  Q1(X1)>2 and  Q1(X2)<1 2and  Q1(V)>2 or  Q1(X1)<1 2and  Q1(X2)>2 and  Q1(V)>2, S∗, if  Q1(X1)<1 2and  Q1(X2)<1 2 or  Q1(X1)>2 and  Q1(X2)<1 2and  Q1(V)<1 2 or  Q1(X1)<1 2and  Q1(X2)>2 and  Q1(V)<1 2, randomly select AS∗or S∗, each with probability 1 2 , otherwise 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 Journal of the Korean Statistical Society (2024) 53:666–703 5 Numerical results 5.1 Robustness We investigate the robustness for various distributions to propose the adaptive test based on the selector statistic. In this paper, we compare the performances of T1 , T2 , T3 , T4 , T(1) max , T(2) max , AD1 and AD2 . We use the exact critical values listed in Table3 when the sample sizes are (n1,n2)=(10, 10),(10, 5) . On the other hand, when the sample sizes are (n1,n2)=(100, 50),(100, 100),(200, 100),(200, 200) , we use the 95% point of the limiting null distribution given in Theorems 1 and 2. In addition, from the results of Kössler and Mukherjee (2020) and Theorems 1 and 2 in this paper, the limiting null distributions of T1 and T3 are same as those of T2 and T4 , respectively. Therefore, we use the asymptotic critical values for T1 , T3 , AD1 and AD2 . For T(1) max and T(2) max , we use the estimated critical values listed in Table1. We show the type-I error rates (5%) for various test statistics in Table 4. Although the competing statistics T1 , T2 , T3 , T4 , T(1) max and T(2) max are distributionfree under the null hypothesis, the independence between the selector and AD1 or AD2 is not guaranteed. Therefore, we investigated the type-I error rates of AD1 and AD2 to ensure they are safely used. The simulated results are based on 1,000,000 replications of Monte-Carlo simulations. Table4 indicates that the type-I errors of statistics are around the significance level as expected because exact critical values are used for small sample sizes. For larger sample sizes, since the type-I error begins to converge, the estimated critical values for some selected sample sizes and the asymptotic critical values are useful. In practice, an approximate permutation test with a random sample is possible for evaluating p value for other sample sizes. From the numerical results, we can also see that the null distribution of T1 and T2 and the null distribution of T3 and T4 are the same. Then, we can select T1 or T2 and T3 or T4 by our selecting rule. We also estimated rejection probabilities when populations deviate from the assumed location-scale-shape family. However, we only comment on the results to save the space. In this paper, we focus on the following pairs of distributions, N(0, 1) with LG (0, √ 3 𝜋 ) , N (0, 3 2) with t6 , U (0, � 3 5 )+5− √15 10 with Be(2, 2) and LG (0, 3 √2𝜋 ) with t6 . In these cases, both populations’ mean, variance and skewness are same but the distributions are different. Therefore, our proposed test may not be suitable if the alternative is not in the location-scale-shape family. But we should also note that the Kolmogorov–Smirnov, the Cramér–von Mises or the Anderson–Darling tests Table 3 The exact and asymptotic critical values ( 𝛼=0.05 ) –, the asymptotic critical value is not 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 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 Table 4 Type-I error of various test statistics for various densities (n1,n2) N(0, 1) GL(0, 1) GR(0, 1) U(0, 1) Exp(2) t2 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 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 Table 4 (continued) (n1,n2) N(0, 1) GL(0, 1) GR(0, 1) U(0, 1) Exp(2) t2 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 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 designed for general alternatives are also not very practical in these cases and future researches on this are highly warranted. 5.2 Power comparison Kössler and Mukherjee (2020) and Mukherjee etal. (2021) compared the powers of T1 and T3 with various existing statistics and they showed the validity of T1 or T3 for various distributions. Therefore, we focus on comparing the power of T1 , T2 , T3 , T4 , T(1) max , T(2) max , AD1 , AD2 , the statistic of Boos (1986), abbreviated by BOOS , the Kolmogorov–Smirnov statistic (Gibbons & Chakraborti, 2021) (KS), the Cramér–von Mises statistic (Anderson, 1962) (CvM), and the Anderson–Darling statistic (Pettitt, 1976) (A–D), for (n1,n2)=(10, 10) and (10,5) in this section. Note that the test statistic of Boos (1986) is a combination of several multisample linear rank statistics. It capitalizes the notion of the Legendre polynomials to construct a multisample statistic. Based on the first three Legendre polynomials, the test statistic of Boos (1986) is a test statistic for location, scale and skew parameters as follows: Table 4 (continued) 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 Journal of the Korean Statistical Society (2024) 53:666–703 There are broadly seven possible types of shifts, three of which are an isolated shift in one of the three parameters, location, scale and shape; another three are mixed shifts involving any two out of three parameters, and a situation with a shift in all the three parameters. We use the normal and the logistic distributions as examples for symmetric distributions. In addition, we use left-skewed and right-skewed Gumbel distributions as examples of asymmetric distributions. The power patterns of five statistics under various alternatives are similar to these four distributions. Therefore, to save space, we only display the results of normal, logistic, left-skewed Gumbel and right-skewed Gumbel distribution. We show the results for the symmetric distributions and the asymmetric distributions in Tables5, 6, 7, 8, respectively. The statistics T1 and T2 (or T3 and T4 ) differ only by the components AS and S designed for deviations in shape parameters. Therefore, there is no difference in the powers of T1 , T2 , AD1 and T(1) max (or T3 , T4 , AD2 and T(2) max ) for the pure location or the pure scale parameter in symmetric distributions with equal sample sizes. From Tables5, 6, 7 and 8, altogether T1 is better than T2 which is in accordance with the theory since test AS is optimal for Lehmann alternatives. Further, in most cases the power of AD1 ( AD2 ) is between that of T1 and T2 ( T3 and T4 ) as expected. Tests based on the Euclidean distance seem to be better than that based on the Mahalanobis distance. The winner of our study is test T(1) max densely followed by T1 . The adaptive test AD1 is on the third place. The other tests are worse. The max-test includes the better of the two tests AS and S for the current situation. The Adaptive test might be considered as a competitor which might be improved by a possibly better selecting rule. Nevertheless, we suggest to apply the Max test T(1) max as it is simpler and better than the adaptive test in many situations. Furthermore, compared to the goodness-of-fit tests, in many cases, the power of T(1) max is greater than or similar to the maximal power of the three tests CvM , KS , and A–D. In fact, we computed the ranks of the statistics according to the simulated power. For each alternative configuration, the worst power is assigned the rank one, that with the largest power is assigned the rank 12. Thus, the test statistics with the largest rank sums are the best. In Table9, we list the rank sums of statistics over all 17 considered alternatives for (n1,n2)=(10, 10) and (10, 5). Table9 shows that T(1) max has the largest rank sum for normal, logistic, left-skewed Gumbel and right-skewed Gumbel distributions. Further, the rank sum of AD1 ( AD2 ) is between that of 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 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 Table 5 Power of various tests for normal and logistic distributions (n1,n2)=(10, 10) with 𝛼=0.05 (𝜇,𝜎,𝛿) T1 T2 T3 T4 AD1 AD2 T(1) max T(2) max BOOS CvM KS A–D The null distribution 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 distribution 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 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 Table 5 (continued) (𝜇,𝜎,𝛿) T1 T2 T3 T4 AD1 AD2 T(1) max T(2) max BOOS CvM 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 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 Table 6 Power of various tests for normal and logistic distributions (n1,n2)=(10, 5) with 𝛼=0.05 (𝜇,𝜎,𝛿) T1 T2 T3 T4 AD1 AD2 T(1) max T(2) max BOOS CvM KS A–D The null distribution 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 distribution 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 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 Table 6 (continued) (𝜇,𝜎,𝛿) T1 T2 T3 T4 AD1 AD2 T(1) max T(2) max BOOS CvM 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 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 Table 10 AST data Hepatitis 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 Cirrhosis 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 Test Statistic W AB AS S L T1 T2 T3 T4 p values 0.0277 0.4120 0.0524 0.1789 0.0622 0.0460 0.0850 0.0770 0.0750 Test Statistic AD1 AD2 T(1) max T(2) max BOOS CvM KS A–D p values 0.0465 0.0770 0.0644 0.1052 0.0546 0.0197 0.0141 0.0279 697 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 Table 11 GGT data Hepatitis 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 Cirrhosis 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 Test Statistic W AB AS S L T1 T2 T3 T4 p values 0.0703 0.5864 0.0597 0.1798 0.1675 0.0928 0.1615 0.3114 0.3115 Test Statistic AD1 AD2 T(1) max T(2) max BOOS CvM KS A–D p values 0.0932 0.3114 0.1236 0.4116 0.2803 0.0769 0.0972 0.1101 698 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 and scale of GGT for hepatitis differ from those of cirrhosis. However, the skewness of GGT for hepatitis is similar to that of cirrhosis. In fact, the skewness of GGT for hepatitis and cirrhosis are 2.304 and 2.223, respectively. This case looks similar to the case of (1, −1, 0) for GR(0, 1) in Table8. Note that the presentation of p values is only for illustration of the considered tests. A clear decision based on all nine tests cannot be drawn. Since we suggested to apply test T(1) max the null hypothesis is rejected for the AST data at the 0.1 level, but for the GGT data not. 7 Concluding remarks For the past 50 years, there have been studies on simultaneously testing the two-sample location and the scale parameters. However, only a few pieces of literature consider simultaneous location, scale, and shape parameter testing. This paper introduced the distribution-free and robust adaptive tests AD1 and AD2 to test for the location, scale, or shape simultaneously. The test proposed by Kössler and Mukherjee (2020) is a combination of the standardized Wilcoxon, the standardized Ansari–Bradley and the standardized Anti-Savage tests. In this paper, we additionally consider test statistics where the Anti-Savage test is replaced with the Savage test. We derive the asymptotic distribution of the proposed tests. In addition, we suggest a selection rule for an adaptive test. Moreover, we considered max-type tests. We investigate the behavior of the power of the tests for small sample sizes via Monte Carlo simulation. In average, the max-type test based on the Euclidean distances is shown to be the best. We also discussed specific data from biomedical experiments. For future research, we may think of combining the Wilcoxon rank-sum test, the Ansari–Bradley test, the Anti-Savage test and the Savage test as the quad-aspect test statistics, as suggested by a referee. Another research topic may be, for example, a simple new study where the Ansari–Bradley test is replaced with another scale test. More generally, other rank scores for location and scales are also worth considering. Appendix1: Proof ofTheorem1 The statistic T2 can be written as a quadratic form of three independently and identically distributed standard normal variables, say Wi , i=1, 2, 3 . Thus T2 = ∑3 i=1 𝜆 i W2 i ∶= Q (say). Let tQ,1−𝛼 and tZ,1−𝛼 be the 1−𝛼 quantiles; 𝜇Q and 𝜇Z be the means; and 𝜎Q and 𝜎Z be the standard deviations of the distribution of Q and Z, respectively. We apply the 𝜒2 approximation proposed by Liu etal. (2009) cf. also Yamaguchi and Murakami (2023). The eigenvalues of 𝚺S are 𝜆1=1.9284, 𝜆2=1, 𝜆3=0.0716. 699 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 The quadratic form Q is approximated by a 𝜒2 distribution suitably shifted and scaled. The degrees of freedom and the location and scale parameters are to be determined. Denote In our case, we have Following Liu etal. (2009), the df of Q are given by and noncentrally parameter by zero. To determine location and scale, we note that Let be expectation and variance of Z, where Z ∼𝜒 2 df and df =1.5786 . For approximation of the quadratic form Q, we shall have Then, the proof is completed. ◻ Appendix2: Proof ofTheorem2 This assertion follows immediately from Lemma 1 and from the decomposition 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, that is, s 2 1 <s 2. df = 1 s2 1 = 1.5786, 𝜇Q =E[Q]=c 1 =3, 𝜎 Q = √ V[Q]= √ 2c 2 = 3.0737. 𝜇Z=E[Z]=df =1.5786, 𝜎Z=√V[Z]=√2df =1.7769 Q−𝜇 Q 𝜎Q ≈Z−𝜇Z 𝜎Z Q≈ 𝜎Q 𝜎Z Z− 𝜎Q 𝜎Z 𝜇Z+𝜇Q =√2c2 √2df Z−√2c2 √2df ⋅df +c 1 =√c2s1Z−√c2 s1 +c1 =c3 c2 Z− c2 2 c3 +c1 ≈1.7299Z+0.2692. 700 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 where Y∼N(0,I) . ◻ Appendix3: Proof ofTheorem3 Recall that T 2=TM 2 T � M2 . Then, we obtain the following lemma by replacing 𝜇j , j=1, 2, 3 in Lemma 4.2 of Kössler and Mukherjee (2020) with 𝜸 given in Lemma 2. Lemma 2 Under the sequence 𝚯 � N =  𝚯�∕ √ N=(𝜇 ,𝜎 , 𝛿)∕ √N and with n1∕ N→𝜆 ∈(0, 1) as min(n1,n2) → ∞ the limiting distribution of T′ M 2 is N( 𝜸 �,𝚺S) , with the asymptotic expectation 𝜸=(𝛾W,𝛾AB,𝛾S)=(𝛾W(f),𝛾AB(f),𝛾S(f)) with Obviously, the asymptotic expectation can be written as Recall 𝜆i are the eigenvalues of 𝚺S in (5). Let 𝚲=diag(𝜆1,𝜆2,𝜆3) , and U be the matrix of eigenvectors of 𝚺S . With transformation we convert the vector T′ M2 to W∼N( 𝚲− 1 2U� 𝜸 � , I ) . Therefore, we have for the expectation Let us introduce 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(f)=− √ 𝜆(1−𝜆)⋅ 𝜇 dW,Location +𝜎 dW, Scale +  𝛿dW, Lehmann √IW , 𝛾 AB =𝛾AB(f)=− √𝜆(1−𝜆)⋅ 𝜇 dAB, Location +𝜎 dAB, Scale + 𝛿dAB, Lehmann √IAB , 𝛾S=𝛾S(f)=− √ 𝜆(1−𝜆)⋅ 𝜇 dS, Location +𝜎 dS, Scale + 𝛿dS, Lehmann √ I S . 𝜸 =− √ 𝜆(1−𝜆)M S (f) 𝚯 . W =𝚲− 1 2U�T� M2 , 𝚫 ∶= ⎛ ⎜ ⎜ ⎝ Δ1 Δ2 Δ3 ⎞ ⎟ ⎟ ⎠ =E[W]=𝚲−1 2U�𝜸�=− √𝜆(1−𝜆)𝚲−1 2U�MS(f) 𝚯 =− √ 𝜆(1−𝜆) ⎛ ⎜ ⎜ ⎝ 0.4750 −0.1835 0.5092 0.3603 0.9328 0 −2.4650 0.9522 2.6425 ⎞ ⎟ ⎟ ⎠ MS(f) ⎛ ⎜ ⎜ ⎝ 𝜇 𝜎  𝛿 ⎞ ⎟ ⎟ ⎠ . 701 1 3 Journal of the Korean Statistical Society (2024) 53:666–703 In addition, define that if s2 1 (𝚫) ≤ s 2 (𝚫 ) , if s2 1 (𝚫)>s 2 (𝚫 ) , ◻ Appendix4: Proof ofTheorem4 Recall that T M2= � W−E[W] √ V[W],AB−E[AB] √ V[AB],S−E[S] √ V[S] � is the vector of the three components of T4 , 𝜼=E[TM2] is its expectation vector, and 𝚺M2 the correlation matrix of this vector. Recall that This statistic is, under the alternative 𝚯 , asymptotically 𝜒2 distributed with three degrees of freedom and noncentrality parameter which can be seen from the decomposition 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) . df (𝚫)= 1 s2 1(𝚫), 𝛽 0(𝚫)=c1(𝚫)−c2 2(𝚫) c3(𝚫) , 𝛽 1(𝚫)=c3(𝚫) c 2( 𝚫 ) , df � (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)−{df �(𝚫)+nc(𝚫)}√c2(𝚫) a(𝚫) , 𝛽 � 1(𝚫)= √ c2(𝚫) a( 𝚫 ) . T 4=TM 2 𝚺 −1 M2 T � M2 . nc (f) ∶= 𝜼𝚺 −1 S 𝜼 � =𝜆(1−𝜆)𝚯 � M S (f) � 𝚺 −1 S (f)M S (f)𝚯 , 702 Journal of the Korean Statistical Society (2024) 53:666–703 1 3 where Y =𝚺 −1∕2 M2 T� M2 and Y ∼N(𝚺 −1∕2 M2 𝜼�,I ) . ◻ Acknowledgements The authors wish to thank the Editor, Associate Editor and two anonymous reviewers for their kind cooperation to improve the article. We appreciate the voluntary contributions of the reviewers affording time to sincerely read the early version of the manuscript. Funding Open Access funding provided by Tokyo University of Science. Data availability The dataset used in this paper is publicly available, with references provided in the text. Declarations Conflict of interest All authors have declared no Conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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