A Generalization of the Sukhatme's Test for Two-Sample Scale Problem

Manish Goyal; Narinder Kumar

doi:10.2991/jsta.d.190524.002

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Volume 18, Issue 2, June 2019, Pages 155 - 170

A Generalization of the Sukhatme's Test for Two-Sample Scale Problem

Authors

Manish Goyal, Narinder Kumar^*

Department of Statistics, Panjab University, Chandigarh, India

^*Corresponding author. Email: nkumar@pu.ac.in

Corresponding Author

Narinder Kumar

Received 17 October 2017, Accepted 1 May 2018, Available Online 4 June 2019.

DOI: 10.2991/jsta.d.190524.002 How to use a DOI?
Keywords: Sukhatme statistics; Subsample; Asymptotic distribution; Pitman efficiency; p-value; Simulation study
Abstract: In this paper, we present nonparametric tests for the two-sample scale problem. The proposed tests include as special case the B.V. Sukhatme, Ann. Math. Stat. 28 (1957), 188–194, and J.V. Deshpande, K. Kusum, Aust. J. Stat. 26 (1984), 16–24 tests. The asymptotic distribution of the test statistics is derived and its Pitman efficiency is worked out with respect to some competing tests. For the illustrative purpose, a numerical example for a real life data set is provided. The simulation study is carried out to assess the power of proposed tests.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Let X1,…,Xn1 and Y1,…,Yn2 be two independent random samples from populations X and Y having absolutely continuous distribution functions (cdfs) Fx and Gx, respectively with common known quantile ξq of order q, that is, Fξq=Gξq=q, 0≤q≤1. Without loss of generality, we assume that ξq is zero for prespecified q. Suppose that the populations X and Y are alike except differing in their scale parameters. Thus, if we take Gx=Fx/θ, then we wish to test the null hypothesis:

H0 : θ=1.

against the alternative hypothesis:

(1)HA : θ>1.

Under the null hypothesis, X′s and Y′s are alike, but under the alternative, Y′s will have more variation than X′s.

For the above problem, with the condition that two distributions have same median, many nonparametric tests are available in literature including Mood [1], Sukhatme [2,3], Ansari and Bradley [4], Siegel and Tukey [5], Capon [6], Klotz [7], Tamura [8–10], Yanagawa [11], Kochar and Gupta [12], Kössler [13,14], Öztürk [15], Kössler and Kumar [16], and references cited therein.

Nonparametric tests for the two-sample scale problem with common quantile different from median was initially proposed by Deshpande and Kusum [17], which was further modified by Kusum [18], Mehra and Rao [19], Mahajan et al. [20], and Kössler and Kumar [21]. This type of problem has several practical applications.

As an example, consider the survey of Hills M. and M345 course team of The Open University, given in Hand et al. [22], in which the survey team asked from two groups of students to guess the width of a lecture hall in the metres for group 1 and in feet for group 2. In this survey, 5% of students guess the width less than the common width for both the groups, which shows that both groups do not have the same median but rather have same quantile of order 0.05. Now, in such a case the testing problem is to check whether the variation in guessing in metres is more than guessing in feet or not, when both groups have common quantile of order 0.05.

The tests are proposed in Section 2 and their distributions are established in Section 3. The comparison of tests with respect to (w.r.t.) some existing tests, in terms of Pitman asymptotic relative efficiencies (AREs) is given in Section 4. To see the implementation of proposed tests, an illustrative example is provided in Section 5. In Section 6, Monte Carlo simulation study is carried out to assess the performance of proposed tests.

2. THE PROPOSED TESTS

Consider m and j as fixed nonnegative integers such that 1≤2m+1≤n1 and 1≤j≤n2. Define the following two kernels h1 and h2 as

h1X1,…,X2m+1;Yj=1,if 0≤MX≤Yj and Xi, Yj≥0;i=1,…, 2m+1 or Yj≤MX≤0 and Xi, Yj≤0;i=1,…, 2m+10, otherwise

h2X1,…,X2m+1;Yj=1,if 0≤MX≤Yj and Xi, Yj≥0;i=1,…, 2m+1 or Yj≤MX≤0 and Xi, Yj≤0;i=1,…, 2m+1 −1, if 0≤Yj≤MX and Xi, Yj≥0;i=1,…, 2m+1 or MX≤Yj≤0 and Xi, Yj≤0;i=1,…, 2m+10, otherwise.

where MX = Median of X1,…,X2m+1.

The two-sample U-statistics associated with kernel h(c),c=1, 2 is defined as

(2)Um(c)X1,…,Xn1;Y1,…,Yn2=n12m+1 n21−1∑s∑j=1n2h(c)Xi1,…,Xi2m+1;Yj,

where s is summation extended over all possible combinations i1,…,i2m+1 of 2m+1 integers chosen from 1,…,n1. The test rejects H0 in favor of HA for large values of Um(c),c=1, 2.

In particular

For m=0, the test statistics Um1 corresponds to test statistics of Sukhatme [2].
For m=0, the test statistics Um2 corresponds to test statistics of Deshpande and Kusum [17].

Thus the proposed U-statistics Um1 and Um2 are the extended version of tests of Sukhatme [2] and Deshpande and Kusum [17], respectively.

3. DISTRIBUTION OF THE TEST STATISTICS

The expected value of Um(c) is

EUm(c)=n12m+1 n21−1∑c∑j=1n2Eh(c)Xi1,…,Xi2m+1;Yj=Eh(c)Xi1,…,Xi2m+1;Yj.

For c=1,

EUm1=∫0∞PMX≤tPYj=tdt+∫−∞0PMX≥tPYj=tdt.

Under H0,

(3)EUm1=121−q2m+2+q2m+2.

For c=2,

EUm2=∫0∞PMX≤tPYj=tdt+∫−∞0PMX≥tPYj=tdt.−∫0∞PMX≥tPYj=tdt−∫−∞0PMX≤tPYj=tdt.

Under H0,

(4)EUm2=0 for all values of m.

The following theorem provides us the asymptotic normality of Um(c) which follows from the well-known theory of U-Statistics (see Lehmann [23]).

Theorem 3.1.

Let N=n1+n2. The asymptotic distribution of N1/2Um(c)−EUm(c) as N→∞ in such a way that that n1/N→λ, 0<λ<1, is normal with mean zero and variance, σ2Um(c), as

(5)σ2Um(c)=2m+12ζ10(c)λ+ζ01(c)1−λ.

Here

ζ10(c)=Eh(c)x,X2,…,X2m+1;Y12−EUm(c)2

and

ζ01(c)=Eh(c)X1,X2,…,X2m+1;y2−EUm(c),

where

h(c)x,X2,…,X2m+1;Y1=E h(c)X1,X2…,X2m+1;Y1|X1=x

and

h(c)X1,X2,…,X2m+1;y=E h(c)X1,X2…,X2m+1;Y1|Y1=y.

Under H0, after some involved computations, we establish the asymptotic null variance, σ02Um(c) as

(6)σ02Um(c)=2m+12ρm(c)λ1−λ,

where for c=1,

ρm1=∑i=m+12m∑j=m+12m2mi2mj∑r=0i∑s=0jirjs −1i+j+r+s2m−r+12m−s+1+22mm∑i=m+12m2mi∑r=0iir−1i+r2m−r+1×∑s=0mms−1m+s2m−s+12m−s+2+2mm2∑r=0m∑s=0mmrms −1r+s2m−r+12m−s+14m−r−s+3×1−q4m+3+q4m+3−EUm12

and for c=2,

ρm2=2mm24∑k=0m∑l=0mmkml−1k+l2m−k+12m−l+14m−k−l+3−4∑l=0mml−1m+l2m−l+12m−l+2×∑k=0mmk−1m+k2m−k+1+∑k=0m∑l=0mmkml−1k+l2m−k+12m−l+1×1−q4m+3+q4m+3.

4. ASYMPTOTIC RELATIVE EFFICIENCY

In this Section, we compare the tests based on Um(c) relative to some existing tests, in the sense of Pitman AREs. The efficacy of the test statistics Um(c) under the sequence of local alternatives, θN=N−1/2θ, is

(7)e2Um(c)=limN→∞ddθEUm(c)|θ=12Nσ02Um(c)=c22m+12mm2σ02Um(c)∫0∞Fy−qm1−Fymf 2ydy−∫−∞0Fymq−Fymf 2ydy2.

Now, we compare the proposed tests based on Um(c) w.r.t. some existing tests for two-sample scale problem, namely, Sukhatme [2] test S, Deshpande and Kusum [17] test DK, Kusum [18] test K, Mahajan et al. [20] test MGA, and some members of Kössler and Kumar [21] test Tk. We also compare proposed tests Um1 and Um2 with each other as well.

The efficacies of S, DK, K, MGA, and Tk tests are

(8)e2S=λ1−λ112−q21−q2∫−∞∞|y| f 2ydy2e2DK=4 λ1−λ13−q1−q∫−∞∞|y| f 2ydy2e2K=112 λ1−λq7+1−q7∫−∞∞|y|Fy−q2f 2ydy2e2MGA=831 600 λ1−λ131q11+1−q11∫−∞0yF2yq−Fy2f 2ydy−∫0∞yFy−q21−Fy2f 2ydy2e2Tk=4k24k−1λ1−λq4k−1+1−q4k−1∫−∞∞|y|Fy−q2k−2f 2ydy2.

In the following Tables 1–10, we have computed the AREs of Um1 and Um2 tests w.r.t. competing tests for some underlying distributions.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um2
	0	1.0000	0.8086	0.3546	1.4079	0.2257	0.1655	0.8086
	1	0.2913	0.2355	0.1033	0.4101	0.0657	0.0482	0.3511
0.1	2	0.1314	0.1062	0.0466	0.1849	0.0296	0.0217	0.1849
	3	0.0730	0.0590	0.0259	0.1028	0.0165	0.0121	0.1122
	4	0.0460	0.0372	0.0163	0.0648	0.0104	0.0076	0.0750
	0	1.0000	0.7506	0.3548	1.4187	0.2274	0.1669	0.7506
	1	0.1806	0.1355	0.0641	0.2562	0.0411	0.0301	0.2178
0.2	2	0.0719	0.0540	0.0255	0.1020	0.0164	0.0120	0.1020
	3	0.0384	0.0288	0.0136	0.0544	0.0087	0.0064	0.0594
	4	0.0237	0.0178	0.0084	0.0337	0.0054	0.0040	0.0390
	0	1.0000	0.7859	0.4105	1.7155	0.2750	0.2037	0.7859
	1	0.1256	0.0987	0.0515	0.2154	0.0345	0.0256	0.1753
0.3	2	0.0421	0.0331	0.0173	0.0723	0.0116	0.0086	0.0722
	3	0.0215	0.0169	0.0088	0.0368	0.0059	0.0044	0.0406
	4	0.0131	0.0103	0.0054	0.0225	0.0036	0.0027	0.0263
	0	1.0000	0.9067	0.4617	2.1225	0.3403	0.2710	0.9067
	1	0.1426	0.1293	0.0658	0.3027	0.0485	0.0386	0.2238
0.4	2	0.0333	0.0302	0.0154	0.0707	0.0113	0.0090	0.0707
	3	0.0137	0.0124	0.0063	0.0291	0.0047	0.0037	0.0345
	4	0.0075	0.0068	0.0035	0.0160	0.0026	0.0020	0.0210
	0	1.0000	1.0000	0.4286	1.7013	0.2727	0.2000	1.0000
	1	0.6863	0.6863	0.2941	1.1675	0.1872	0.1373	1.0000
0.5	2	0.5878	0.5878	0.2519	1.0000	0.1603	0.1176	1.0000
	3	0.5383	0.5383	0.2307	0.9158	0.1468	0.1077	1.0000
	4	0.5079	0.5079	0.2177	0.8641	0.1385	0.1016	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 1

AREs of Um1 w.r.t. competing tests for uniform distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um2
	0	1.0000	0.8086	0.5867	0.9251	0.5485	0.5452	0.8086
	1	0.3977	0.3216	0.2333	0.3679	0.2181	0.2168	0.3511
0.1	2	0.1999	0.1616	0.1173	0.1849	0.1096	0.1090	0.1849
	3	0.1174	0.0950	0.0689	0.1086	0.0644	0.0640	0.1122
	4	0.0766	0.0619	0.0449	0.0708	0.0420	0.0417	0.0750
	0	1.0000	0.7506	0.5693	0.9229	0.5292	0.5221	0.7506
	1	0.2496	0.1873	0.1421	0.2303	0.1321	0.1303	0.2178
0.2	2	0.1106	0.0830	0.0629	0.1020	0.0585	0.0577	0.1020
	3	0.0622	0.0467	0.0354	0.0574	0.0329	0.0325	0.0594
	4	0.0397	0.0298	0.0226	0.0367	0.0210	0.0207	0.0390
	0	1.0000	0.7859	0.6525	1.0998	0.6272	0.6217	0.7859
	1	0.1751	0.1376	0.1143	0.1926	0.1099	0.1089	0.1753
0.3	2	0.0656	0.0516	0.0428	0.0722	0.0412	0.0408	0.0722
	3	0.0353	0.0278	0.0231	0.0389	0.0222	0.0220	0.0406
	4	0.0222	0.0175	0.0145	0.0245	0.0139	0.0138	0.0263
	0	1.0000	0.9067	0.7546	1.3814	0.7938	0.8398	0.9067
	1	0.1951	0.1769	0.1472	0.2695	0.1549	0.1638	0.2238
0.4	2	0.0512	0.0464	0.0386	0.0707	0.0406	0.0430	0.0707
	3	0.0224	0.0203	0.0169	0.0309	0.0178	0.0188	0.0345
	4	0.0128	0.0116	0.0096	0.0176	0.0101	0.0107	0.0210
	0	1.0000	1.0000	0.7232	1.1531	0.6690	0.6577	1.0000
	1	0.9151	0.9151	0.6618	1.0552	0.6122	0.6018	1.0000
0.5	2	0.8672	0.8672	0.6272	1.0000	0.5802	0.5704	1.0000
	3	0.8360	0.8360	0.6046	0.9640	0.5593	0.5499	1.0000
	4	0.8137	0.8137	0.5885	0.9382	0.5443	0.5351	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 2

AREs of Um1 w.r.t. competing tests for normal distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um2
	0	1.0000	0.8086	0.6438	0.8756	0.6475	0.6812	0.8086
	1	0.4129	0.3339	0.2658	0.3615	0.2674	0.2813	0.3511
0.1	2	0.2112	0.1707	0.1360	0.1849	0.1367	0.1439	0.1849
	3	0.1253	0.1013	0.0807	0.1097	0.0812	0.0854	0.1122
	4	0.0823	0.0665	0.0530	0.0720	0.0533	0.0561	0.0750
	0	1.0000	0.7506	0.6201	0.8753	0.6161	0.6396	0.7506
	1	0.2590	0.1944	0.1606	0.2267	0.1596	0.1656	0.2178
0.2	2	0.1166	0.0875	0.0723	0.1021	0.0719	0.0746	0.1020
	3	0.0662	0.0497	0.0411	0.0580	0.0408	0.0424	0.0594
	4	0.0425	0.0319	0.0264	0.0372	0.0262	0.0272	0.0390
	0	1.0000	0.7859	0.7073	1.0442	0.7229	0.7511	0.7859
	1	0.1816	0.1427	0.1285	0.1896	0.1313	0.1364	0.1753
0.3	2	0.0692	0.0543	0.0489	0.0722	0.0500	0.0519	0.0722
	3	0.0376	0.0295	0.0266	0.0392	0.0272	0.0282	0.0406
	4	0.0238	0.0187	0.0168	0.0248	0.0172	0.0179	0.0263
	0	1.0000	0.9067	0.8179	1.3189	0.9106	1.0060	0.9067
	1	0.2012	0.1824	0.1645	0.2653	0.1832	0.2024	0.2238
0.4	2	0.0536	0.0486	0.0438	0.0707	0.0488	0.0539	0.0707
	3	0.0236	0.0214	0.0193	0.0312	0.0215	0.0238	0.0345
	4	0.0135	0.0123	0.0111	0.0178	0.0123	0.0136	0.0210
	0	1.0000	1.0000	0.7859	1.1109	0.7688	0.7877	1.0000
	1	0.9385	0.9385	0.7375	1.0425	0.7214	0.7392	1.0000
0.5	2	0.9002	0.9002	0.7074	1.0000	0.6920	0.7091	1.0000
	3	0.8738	0.8738	0.6867	0.9707	0.6717	0.6883	1.0000
	4	0.8542	0.8542	0.6713	0.9490	0.6567	0.6729	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 3

AREs of Um1 w.r.t. competing tests for logistic distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um2
	0	1.0000	0.8086	0.7937	0.7794	0.9290	1.0704	0.8086
	1	0.4436	0.3587	0.3521	0.3458	0.4121	0.4748	0.3511
0.1	2	0.2373	0.1918	0.1883	0.1849	0.2204	0.2539	0.1849
	3	0.1453	0.1175	0.1153	0.1133	0.1350	0.1555	0.1122
	4	0.0977	0.0790	0.0775	0.0761	0.0907	0.1045	0.0750
	0	1.0000	0.7506	0.7651	0.8107	0.8528	0.9374	0.7506
	1	0.2719	0.2041	0.2080	0.2204	0.2319	0.2549	0.2178
0.2	2	0.1259	0.0945	0.0963	0.1021	0.1074	0.1180	0.1020
	3	0.0728	0.0546	0.0557	0.0590	0.0621	0.0682	0.0594
	4	0.0474	0.0356	0.0362	0.0384	0.0404	0.0444	0.0390
	0	1.0000	0.7859	0.8434	1.0247	0.9259	0.9890	0.7859
	1	0.1839	0.1445	0.1551	0.1884	0.1703	0.1818	0.1753
0.3	2	0.0705	0.0554	0.0595	0.0722	0.0653	0.0697	0.0722
	3	0.0384	0.0302	0.0324	0.0394	0.0356	0.0380	0.0406
	4	0.0243	0.0191	0.0205	0.0249	0.0225	0.0241	0.0263
	0	1.0000	0.9067	0.9080	1.3571	1.0485	1.1730	0.9067
	1	0.1978	0.1793	0.1796	0.2684	0.2074	0.2320	0.2238
0.4	2	0.0521	0.0472	0.0473	0.0706	0.0546	0.0611	0.0707
	3	0.0227	0.0206	0.0207	0.0309	0.0239	0.0267	0.0345
	4	0.0130	0.0118	0.0118	0.0176	0.0136	0.0152	0.0210
	0	1.0000	1.0000	0.8216	1.1185	0.8107	0.8302	1.0000
	1	0.9341	0.9341	0.7675	1.0448	0.7573	0.7755	1.0000
0.5	2	0.8941	0.8941	0.7347	1.0000	0.7249	0.7423	1.0000
	3	0.8669	0.8669	0.7123	0.9696	0.7028	0.7197	1.0000
	4	0.8469	0.8469	0.6959	0.9473	0.6866	0.7031	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 4

AREs of Um1 w.r.t. competing tests for Laplace distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um2
	0	1.0000	0.8086	1.0503	0.6981	1.6322	2.4497	0.8086
	1	0.4798	0.3879	0.5039	0.3349	0.7831	1.1753	0.3511
0.1	2	0.2649	0.2142	0.2782	0.1849	0.4324	0.6489	0.1849
	3	0.1649	0.1334	0.1732	0.1151	0.2692	0.4040	0.1122
	4	0.1118	0.0904	0.1175	0.0781	0.1826	0.2740	0.0750
	0	1.0000	0.7506	0.9851	0.7099	1.4326	2.0008	0.7506
	1	0.2991	0.2245	0.2947	0.2123	0.4285	0.5985	0.2178
0.2	2	0.1438	0.1079	0.1416	0.1021	0.2059	0.2876	0.1020
	3	0.0849	0.0637	0.0837	0.0603	0.1217	0.1699	0.0594
	4	0.0559	0.0420	0.0551	0.0397	0.0801	0.1119	0.0390
	0	1.0000	0.7859	1.0879	0.8547	1.5550	2.0871	0.7859
	1	0.2086	0.1639	0.2270	0.1783	0.3244	0.4354	0.1753
0.3	2	0.0845	0.0664	0.0919	0.0722	0.1314	0.1763	0.0722
	3	0.0476	0.0374	0.0517	0.0406	0.0740	0.0993	0.0406
	4	0.0308	0.0242	0.0335	0.0263	0.0479	0.0643	0.0263
	0	1.0000	0.9067	1.2479	1.1145	1.8715	2.5871	0.9067
	1	0.2243	0.2034	0.2800	0.2500	0.4199	0.5804	0.2238
0.4	2	0.0634	0.0575	0.0791	0.0706	0.1186	0.1639	0.0707
	3	0.0289	0.0262	0.0361	0.0322	0.0541	0.0748	0.0345
	4	0.0169	0.0153	0.0211	0.0189	0.0317	0.0438	0.0210
	0	1.0000	1.0000	1.2118	0.9881	1.5808	1.9958	1.0000
	1	1.0145	1.0145	1.2293	1.0023	1.6036	2.0246	1.0000
0.5	2	1.0122	1.0122	1.2266	1.0000	1.6001	2.0201	1.0000
	3	1.0062	1.0062	1.2192	0.9941	1.5905	2.0079	1.0000
	4	0.9995	0.9995	1.2111	0.9875	1.5799	1.9946	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 5

AREs of Um1 w.r.t. competing tests for Cauchy distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um1
	0	1.2367	1.0000	0.4385	1.7412	0.2791	0.2047	1.2367
	1	0.8295	0.6708	0.2941	1.1679	0.1872	0.1373	2.8480
0.1	2	0.7103	0.5743	0.2518	1.0000	0.1603	0.1176	5.4072
	3	0.6504	0.5259	0.2306	0.9157	0.1468	0.1077	8.9105
	4	0.6138	0.4963	0.2176	0.8641	0.1385	0.1016	13.3405
	0	1.3323	1.0000	0.4727	1.8901	0.3030	0.2223	1.3323
	1	0.8290	0.6222	0.2941	1.1761	0.1885	0.1383	4.5904
0.2	2	0.7049	0.5291	0.2501	1.0000	0.1603	0.1176	9.8014
	3	0.6452	0.4843	0.2289	0.9153	0.1467	0.1077	16.8216
	4	0.6088	0.4569	0.2160	0.8637	0.1385	0.1016	25.6632
	0	1.2724	1.0000	0.5223	2.1828	0.3499	0.2592	1.2724
	1	0.7166	0.5631	0.2941	1.2292	0.1970	0.1460	5.7058
0.3	2	0.5829	0.4581	0.2393	1.0000	0.1603	0.1187	13.8383
	3	0.5285	0.4154	0.2169	0.9066	0.1453	0.1077	24.6283
	4	0.4978	0.3912	0.2043	0.8540	0.1369	0.1014	37.9751
	0	1.1029	1.0000	0.5092	2.3409	0.3753	0.2989	1.1029
	1	0.6371	0.5777	0.2941	1.3523	0.2168	0.1726	4.4677
0.4	2	0.4711	0.4272	0.2175	1.0000	0.1603	0.1277	14.1440
	3	0.3973	0.3602	0.1834	0.8432	0.1352	0.1077	28.9889
	4	0.3600	0.3265	0.1662	0.7642	0.1225	0.0976	47.7677
	0	1.0000	1.0000	0.4286	1.7013	0.2727	0.2000	1.0000
	1	0.6863	0.6863	0.2941	1.1675	0.1872	0.1373	1.0000
0.5	2	0.5878	0.5878	0.2519	1.0000	0.1603	0.1176	1.0000
	3	0.5383	0.5383	0.2307	0.9158	0.1468	0.1077	1.0000
	4	0.5079	0.5079	0.2177	0.8641	0.1385	0.1016	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 6

AREs of Um2 w.r.t. competing tests for uniform distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um1
	0	1.2367	1.0000	0.7255	1.1441	0.6783	0.6742	1.2367
	1	1.1325	0.9158	0.6644	1.0477	0.6212	0.6175	2.8480
0.1	2	1.0809	0.8740	0.6341	1.0000	0.5929	0.5893	5.4072
	3	1.0466	0.8463	0.6140	0.9682	0.5740	0.5706	8.9105
	4	1.0215	0.8260	0.5993	0.9450	0.5603	0.5569	13.3405
	0	1.3323	1.0000	0.7585	1.2296	0.7051	0.6955	1.3323
	1	1.1456	0.8599	0.6522	1.0573	0.6063	0.5981	4.5904
0.2	2	1.0836	0.8133	0.6169	1.0000	0.5734	0.5657	9.8014
	3	1.0462	0.7853	0.5956	0.9656	0.5537	0.5462	16.8216
	4	1.0196	0.7653	0.5805	0.9410	0.5396	0.5323	25.6632
	0	1.2724	1.0000	0.8303	1.3995	0.7982	0.7911	1.2724
	1	0.9995	0.7855	0.6522	1.0993	0.6270	0.6214	5.7058
0.3	2	0.9092	0.7146	0.5933	1.0000	0.5704	0.5653	13.8383
	3	0.8699	0.6837	0.5677	0.9568	0.5457	0.5409	24.6283
	4	0.8458	0.6647	0.5519	0.9303	0.5306	0.5259	37.9751
	0	1.1029	1.0000	0.8323	1.5236	0.8755	0.9262	1.1029
	1	0.8714	0.7902	0.6576	1.2039	0.6917	0.7318	4.4677
0.4	2	0.7239	0.6564	0.5462	1.0000	0.5746	0.6079	14.1440
	3	0.6475	0.5871	0.4886	0.8945	0.5140	0.5438	28.9889
	4	0.6075	0.5508	0.4584	0.8392	0.4822	0.5101	47.7677
	0	1.0000	1.0000	0.7232	1.1531	0.6690	0.6577	1.0000
	1	0.9151	0.9151	0.6618	1.0552	0.6122	0.6018	1.0000
0.5	2	0.8672	0.8672	0.6272	1.0000	0.5802	0.5704	1.0000
	3	0.8360	0.8360	0.6046	0.9640	0.5593	0.5499	1.0000
	4	0.8137	0.8137	0.5885	0.9382	0.5443	0.5351	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 7

AREs of Um2 w.r.t. competing tests for normal distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um1
	0	1.2367	1.0000	0.7963	1.0829	0.8008	0.8425	1.2367
	1	1.1759	0.9508	0.7571	1.0296	0.7614	0.8011	2.8480
0.1	2	1.1421	0.9235	0.7354	1.0000	0.7396	0.7781	5.4072
	3	1.1171	0.9033	0.7192	0.9781	0.7234	0.7610	8.9105
	4	1.0977	0.8876	0.7068	0.9611	0.7108	0.7478	13.3405
	0	1.3323	1.0000	0.8262	1.1661	0.8208	0.8521	1.3323
	1	1.1888	0.8923	0.7372	1.0405	0.7324	0.7603	4.5904
0.2	2	1.1425	0.8576	0.7085	1.0000	0.7039	0.7307	9.8014
	3	1.1134	0.8357	0.6905	0.9745	0.6859	0.7121	16.8216
	4	1.0915	0.8192	0.6769	0.9553	0.6724	0.6981	25.6632
	0	1.2724	1.0000	0.9001	1.3287	0.9199	0.9558	1.2724
	1	1.0364	0.8145	0.7331	1.0821	0.7492	0.7784	5.7058
0.3	2	0.9577	0.7526	0.6774	1.0000	0.6923	0.7193	13.8383
	3	0.9242	0.7264	0.6538	0.9651	0.6681	0.6942	24.6283
	4	0.9035	0.7101	0.6391	0.9434	0.6531	0.6786	37.9751
	0	1.1029	1.0000	0.9021	1.4546	1.0042	1.1095	1.1029
	1	0.8988	0.8150	0.7351	1.1855	0.8184	0.9042	4.4677
0.4	2	0.7582	0.6875	0.6202	1.0000	0.6904	0.7628	14.1440
	3	0.6843	0.6205	0.5597	0.9025	0.6231	0.6884	28.9889
	4	0.6456	0.5854	0.5280	0.8515	0.5878	0.6495	47.7677
	0	1.0000	1.0000	0.7859	1.1109	0.7688	0.7877	1.0000
	1	0.9385	0.9385	0.7375	1.0425	0.7214	0.7392	1.0000
0.5	2	0.9002	0.9002	0.7074	1.0000	0.6920	0.7091	1.0000
	3	0.8738	0.8738	0.6867	0.9707	0.6717	0.6883	1.0000
	4	0.8542	0.8542	0.6713	0.9490	0.6567	0.6729	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 8

AREs of Um2 w.r.t. competing tests for logistic distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um1
	0	1.2367	1.0000	0.9816	0.9639	1.1489	1.3237	1.2367
	1	1.2634	1.0216	1.0028	0.9847	1.1736	1.3523	2.8480
0.1	2	1.2830	1.0375	1.0184	1.0000	1.1919	1.3733	5.4072
	3	1.2951	1.0472	1.0279	1.0094	1.2031	1.3862	8.9105
	4	1.3031	1.0536	1.0342	1.0156	1.2105	1.3947	13.3405
	0	1.3323	1.0000	1.0193	1.0800	1.1361	1.2489	1.3323
	1	1.2484	0.9370	0.9551	1.0120	1.0646	1.1702	4.5904
0.2	2	1.2336	0.9259	0.9438	1.0000	1.0520	1.1564	9.8014
	3	1.2241	0.9188	0.9366	0.9923	1.0439	1.1475	16.8216
	4	1.2154	0.9123	0.9299	0.9853	1.0365	1.1394	25.6632
	0	1.2724	1.0000	1.0731	1.3039	1.1782	1.2583	1.2724
	1	1.0490	0.8244	0.8846	1.0749	0.9712	1.0373	5.7058
0.3	2	0.9759	0.7669	0.8230	1.0000	0.9035	0.9650	13.8383
	3	0.9447	0.7424	0.7967	0.9680	0.8747	0.9342	24.6283
	4	0.9245	0.7266	0.7797	0.9474	0.8560	0.9143	37.9751
	0	1.1029	1.0000	1.0014	1.4967	1.1564	1.2937	1.1029
	1	0.8835	0.8011	0.8022	1.1989	0.9264	1.0363	4.4677
0.4	2	0.7369	0.6682	0.6691	1.0000	0.7727	0.8644	14.1440
	3	0.6604	0.5988	0.5997	0.8963	0.6925	0.7747	28.9889
	4	0.6204	0.5625	0.5633	0.8419	0.6505	0.7277	47.7677
	0	1.0000	1.0000	0.8216	1.1185	0.8107	0.8302	1.0000
	1	0.9341	0.9341	0.7675	1.0448	0.7573	0.7755	1.0000
0.5	2	0.8941	0.8941	0.7347	1.0000	0.7249	0.7423	1.0000
	3	0.8669	0.8669	0.7123	0.9696	0.7028	0.7197	1.0000
	4	0.8469	0.8469	0.6959	0.9473	0.6866	0.7031	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 9

AREs of Um2 w.r.t. competing tests for Laplace distribution.

	Test
q or 1−q	m	S	DK or T1	K or T2	MGA	T3	T4	Um1
	0	1.2367	1.0000	1.2990	0.8633	2.0186	3.0296	1.2367
	1	1.3663	1.1048	1.4351	0.9537	2.2301	3.3470	2.8480
0.1	2	1.4326	1.1584	1.5047	1.0000	2.3382	3.5094	5.4072
	3	1.4698	1.1885	1.5438	1.0260	2.3990	3.6006	8.9105
	4	1.4924	1.2067	1.5675	1.0418	2.4359	3.6559	13.3405
	0	1.3323	1.0000	1.3124	0.9458	1.9086	2.6656	1.3323
	1	1.3732	1.0307	1.3527	0.9748	1.9672	2.7475	4.5904
0.2	2	1.4087	1.0574	1.3877	1.0000	2.0180	2.8185	9.8014
	3	1.4281	1.0719	1.4068	1.0138	2.0458	2.8573	16.8216
	4	1.4383	1.0796	1.4169	1.0210	2.0605	2.8778	25.6632
	0	1.2724	1.0000	1.3843	1.0875	1.9786	2.6557	1.2724
	1	1.1904	0.9356	1.2951	1.0175	1.8511	2.4845	5.7058
0.3	2	1.1701	0.9196	1.2729	1.0000	1.8194	2.4420	13.8383
	3	1.1688	0.9185	1.2715	0.9989	1.8174	2.4393	24.6283
	4	1.1687	0.9185	1.2715	0.9989	1.8173	2.4392	37.9751
	0	1.1029	1.0000	1.3765	1.2293	2.0642	2.8535	1.1029
	1	1.0022	0.9087	1.2508	1.1170	1.8757	2.5930	4.4677
0.4	2	0.8971	0.8135	1.1197	1.0000	1.6791	2.3212	14.1440
	3	0.8387	0.7605	1.0468	0.9349	1.5698	2.1700	28.9889
	4	0.8092	0.7338	1.0100	0.9020	1.5146	2.0937	47.7677
	0	1.0000	1.0000	1.2118	0.9881	1.5808	1.9958	1.0000
	1	1.0145	1.0145	1.2293	1.0023	1.6036	2.0246	1.0000
0.5	2	1.0122	1.0122	1.2266	1.0000	1.6001	2.0201	1.0000
	3	1.0062	1.0062	1.2192	0.9941	1.5905	2.0079	1.0000
	4	0.9995	0.9995	1.2111	0.9875	1.5799	1.9946	1.0000

ARE: Asymptotic relative efficiencies; DK: Deshpande and Kusum test; MGA: Mahajan et al. test; w.r.t: With respect to.

Table 10

AREs of Um2 w.r.t. competing tests for Cauchy distribution.

From the ARE tables, we observe the following:

For light-tailed distributions, like uniform distribution, Um1 and Um2 tests perform as good as or better than S and MGA tests for some specific choices of m and q. However the optimal choice of m is
1. For q=0.5, Um1 and Um2 tests are more efficient than MGA test for m≤2 with maximum efficiency achieved at m=0.
2. For q≠0.5, Um2 test is more efficient than S test for m=0 and is more efficient than MGA test for m≤2 with maximum efficiency achieved at m=0.
For medium-tailed distributions, like normal distribution, Um1 and Um2 tests perform as good as or better than S and MGA tests for some specific choices of m and q. However the optimal choice of m is
1. For q∈0.3, 0.7, then Um1 and Um2 tests are more efficient than S and MGA test for m=0.
2. For q∉0.3, 0.7, Um2 test is more efficient than S and MGA tests for m≤2, with maximum efficiency achieved at m=0.
For large tail distributions, like Cauchy distribution, Um1 and Um2 tests performs better than its competing tests. The optimal choice of m is
1. For q=0.5, Um1 and Um2 tests are more efficient than the competing tests for m≤2 with maximum efficiency achieved at m=1.
2. For q≠0.5, Um2 tests are more efficient than its competing tests with maximum efficiency is for m as large as possible for q∉0.3, 0.7, otherwise the maximum efficiency for Um2 tests is achieved at m=0.
ARE of Um2 test w.r.t. Um1 test doesn't depend upon the underlying distribution. Moreover, the Um2 test is asymptotically equivalent to Um1 test for q=0.5 and Um2 test is always more efficient than Um1 test for q≠0.5. Thus, in general, one should use Um2 test in comparison to Um1 test to gain more efficiency.

5. AN ILLUSTRATIVE EXAMPLE

To see the execution of the tests based on Um(c), we consider the data of the survey of Hills M. and M345 course team of The Open University, given in Hand et al. [22]. In this experiment, two groups of 44 and 69 students were asked to guess the width of a lecture hall in metres and feet, respectively. It is of relevance to check whether there is greater variation in guessing the width in metres in comparison to guessing the width in feet.

By using Kolmogorov–Smirnov test, we have seen that the data set follows Cauchy distribution at 5% level of significance and has common quantile of order 0.05, that is, q=0.05. Therefore for Um2 test, by using the observation 3, made in Section 4, one should consider m as large as possible to have maximum gain in efficiency in comparison to competing tests.

The values of computed Um1 and Um2 tests statistics, and the competing test statistics along with their p-values are given in Table 11.

Test	S	DK or T1	K or T2	MGA	T3	T4
Test statistics	0.536399	0.189655	0.348273	0.248532	0.425221	0.464311
p-value	0.163061	0.118561	0.033079	0.112129	0.018887	0.013862

Test	Um1			Um2
m	0	1	2	0	1	2
Test statistics	0.536399	0.467569	0.400347	0.189655	0.129979	0.066549
p-value	0.163061	0.309472	0.414866	0.118561	0.012831	0.011439

Table 11

Computed values of test statistics and p-values for different tests.

We note that at 5% level of significance, the null hypothesis of same variability in guessing the width in metres in comparison to guessing the width in feet is rejected by tests K, T2, T3, T4, and Um2m=1, 2. However, for tests S, DK, T1, MGA, Um1 m=0, 1, 2, and Um2m=0 tests the null hypothesis is not rejected.

6. SIMULATION STUDY

In this section, using Monte Carlo simulation technique, we have computed the estimated power of Um1 and Um2 tests for sample size n1 and n2 with n1, n2=10, 15, 20,25, 30,40. The computation of power is based on 10,000 repetitions, by generating the data from three common distributions, namely, (i) uniform, (ii) normal, and (iii) Cauchy. The scale parameters considered are θ = 1.5 0.5 3 and level of significance is fixed at 5%.

The idea behind the selecting these three distributions, is that the uniform, normal, and Cauchy have short, medium, and heavy tail, respectively. So it is of relevance to see the test performance for these distributions in terms of power.

The estimated powers are given in Tables 12–17.

q or 1−q		0.1			0.3			0.5
	m
n1, n2	θ	0	1	2	0	1	2	0	1	2
	1.5	0.2377	0.1446	0.1060	0.2465	0.1029	0.0516	0.2643	0.2199	0.1735
	2	0.3466	0.2615	0.2299	0.3512	0.2267	0.1413	0.3718	0.3267	0.3140
10, 10	2.5	0.4754	0.3726	0.3157	0.4876	0.3145	0.1967	0.5079	0.4614	0.4497
	3	0.4992	0.4112	0.3760	0.5017	0.3710	0.2718	0.6220	0.4825	0.4750
	1.5	0.2556	0.1653	0.1298	0.2678	0.1250	0.0829	0.2798	0.2317	0.2252
15, 10	2	0.3679	0.2877	0.2520	0.3910	0.2478	0.1665	0.4029	0.3675	0.3538
	2.5	0.4821	0.4013	0.3517	0.5231	0.3476	0.2115	0.5367	0.5021	0.4694
	3	0.5377	0.4524	0.4064	0.5423	0.4013	0.2995	0.6718	0.5319	0.5168
	1.5	0.2713	0.1861	0.1492	0.2814	0.1448	0.1033	0.2940	0.2528	0.2466
15, 15	2	0.3912	0.3093	0.2816	0.4355	0.2789	0.1834	0.4412	0.4120	0.3975
	2.5	0.5113	0.4415	0.3902	0.5499	0.3850	0.2365	0.5671	0.5317	0.5120
	3	0.5701	0.4836	0.4380	0.5797	0.4320	0.3218	0.7374	0.5690	0.5498
	1.5	0.3055	0.2076	0.1709	0.3244	0.1653	0.1345	0.3383	0.2936	0.2855
20, 15	2	0.4323	0.3517	0.3084	0.4676	0.3034	0.2069	0.4855	0.4534	0.4412
	2.5	0.5435	0.4690	0.4125	0.5892	0.4097	0.2578	0.6054	0.5630	0.5427
	3	0.6211	0.5019	0.4584	0.6283	0.4525	0.3629	0.7990	0.5987	0.5811
	1.5	0.3397	0.2375	0.1967	0.3421	0.1899	0.1656	0.3622	0.3210	0.3118
20, 20	2	0.4650	0.3819	0.3419	0.5116	0.3364	0.2246	0.5249	0.4921	0.4720
	2.5	0.5783	0.4926	0.4520	0.6311	0.4465	0.2790	0.6420	0.6112	0.6009
	3	0.6608	0.5420	0.4879	0.6691	0.4819	0.3935	0.8323	0.6317	0.6202
	1.5	0.3694	0.2542	0.2122	0.3980	0.2067	0.1872	0.4047	0.3618	0.3447
25, 20	2	0.4976	0.4315	0.3601	0.5567	0.3512	0.2457	0.5685	0.5431	0.5230
	2.5	0.6062	0.5436	0.5011	0.6638	0.4951	0.3010	0.6828	0.6642	0.6435
	3	0.6811	0.5994	0.5114	0.7298	0.5078	0.4225	0.8693	0.6913	0.6513
	1.5	0.4009	0.3019	0.2610	0.4330	0.2518	0.2091	0.4428	0.4029	0.3892
25, 25	2	0.5272	0.4628	0.4025	0.5906	0.3970	0.2679	0.6019	0.5772	0.5699
	2.5	0.6351	0.6013	0.5510	0.7110	0.5412	0.3315	0.7292	0.7018	0.6910
	3	0.7215	0.6519	0.5721	0.7820	0.5699	0.4576	0.9086	0.7114	0.7005
	1.5	0.4464	0.3517	0.3167	0.4765	0.3035	0.2335	0.4835	0.4667	0.4487
30, 30	2	0.5511	0.5029	0.4398	0.6514	0.4305	0.2984	0.6626	0.6220	0.6165
	2.5	0.6535	0.6520	0.5999	0.7688	0.5921	0.3629	0.7709	0.7517	0.7374
	3	0.7914	0.7013	0.6310	0.8610	0.6223	0.4898	0.9389	0.7690	0.7589
	1.5	0.4938	0.4044	0.3759	0.5149	0.3649	0.2639	0.5250	0.4990	0.4814
40, 40	2	0.5906	0.5514	0.4895	0.7170	0.4819	0.3247	0.7278	0.6814	0.6678
	2.5	0.7036	0.7130	0.6606	0.8125	0.6566	0.3945	0.8311	0.8155	0.8006
	3	0.8688	0.7412	0.6975	0.9456	0.6836	0.5476	0.9652	0.8432	0.8399

Table 12

Estimated power of Um1 for uniform distribution.

q or 1−q		0.1			0.3			0.5
	m
n1, n2	θ	0	1	2	0	1	2	0	1	2
	1.5	0.3923	0.2295	0.1917	0.1868	0.1135	0.0758	0.2571	0.2103	0.2176
10, 10	2	0.5539	0.2892	0.2503	0.3992	0.1527	0.1079	0.4983	0.3006	0.2824
	2.5	0.6802	0.3207	0.2913	0.5606	0.1816	0.1322	0.6520	0.3802	0.3036
	3	0.7717	0.3961	0.3263	0.7412	0.2084	0.1624	0.7506	0.4712	0.3238
	1.5	0.4407	0.2531	0.2802	0.1973	0.1442	0.1108	0.2861	0.2421	0.2284
15, 10	2	0.6203	0.3265	0.2779	0.4561	0.1778	0.1445	0.5228	0.3445	0.2903
	2.5	0.7209	0.3876	0.3383	0.6284	0.1966	0.1621	0.6849	0.4169	0.3264
	3	0.8319	0.4161	0.3691	0.7783	0.2111	0.1909	0.7829	0.4701	0.4232
	1.5	0.4712	0.2822	0.2561	0.2376	0.1655	0.1427	0.3213	0.2690	0.2316
15, 15	2	0.5998	0.3670	0.3492	0.5147	0.1908	0.1799	0.6285	0.3739	0.3167
	2.5	0.7716	0.4314	0.3614	0.6813	0.2123	0.2013	0.7906	0.4545	0.3433
	3	0.9064	0.5029	0.4553	0.8718	0.2304	0.2177	0.8878	0.5254	0.7545
	1.5	0.5001	0.3168	0.2877	0.2924	0.1885	0.1613	0.3626	0.2803	0.2520
20, 15	2	0.6312	0.4028	0.3789	0.5598	0.2103	0.1981	0.6689	0.4452	0.3469
	2.5	0.8128	0.4733	0.4622	0.7502	0.2292	0.2104	0.8281	0.5674	0.3902
	3	0.9155	0.5614	0.5033	0.9040	0.2586	0.2336	0.9197	0.6591	0.5508
	1.5	0.5233	0.3527	0.3219	0.3362	0.2067	0.1906	0.3881	0.2992	0.2751
20, 20	2	0.6778	0.4556	0.4177	0.6190	0.2214	0.2115	0.7088	0.4718	0.3972
	2.5	0.8602	0.5141	0.4834	0.8123	0.2409	0.2256	0.8853	0.6190	0.4713
	3	0.9330	0.6233	0.5646	0.9536	0.2760	0.2569	0.9549	0.7105	0.6243
	1.5	0.5713	0.3967	0.3536	0.3818	0.2412	0.2278	0.4286	0.3235	0.3038
25, 20	2	0.6966	0.4920	0.4562	0.6645	0.2723	0.2416	0.7611	0.5474	0.4589
	2.5	0.8870	0.5672	0.5251	0.8605	0.2916	0.2550	0.9109	0.7105	0.5472
	3	0.9424	0.6713	0.6109	0.9661	0.3278	0.2714	0.9688	0.7979	0.7094
	1.5	0.6051	0.4535	0.4078	0.4202	0.2813	0.2589	0.4638	0.3377	0.3165
25, 25	2	0.7203	0.5687	0.5087	0.7127	0.3225	0.2726	0.8083	0.5742	0.4991
	2.5	0.9013	0.6079	0.5566	0.9008	0.3609	0.3018	0.9404	0.7448	0.6393
	3	0.9516	0.7123	0.6643	0.9743	0.4035	0.3537	0.9775	0.8564	0.7739
	1.5	0.6768	0.5191	0.4519	0.4612	0.3562	0.3109	0.5116	0.3714	0.3506
30, 30	2	0.7425	0.6286	0.5628	0.7794	0.3843	0.3217	0.8663	0.6413	0.5624
	2.5	0.9309	0.7008	0.6234	0.9389	0.4354	0.3663	0.9703	0.8005	0.6955
	3	0.9731	0.8013	0.7256	0.9822	0.4912	0.4108	0.9873	0.8976	0.8211
	1.5	0.6908	0.6124	0.5327	0.5277	0.4729	0.4019	0.5804	0.5083	0.4139
40, 40	2	0.7612	0.7075	0.6560	0.8513	0.5296	0.4386	0.9024	0.8159	0.6528
	2.5	0.9489	0.7716	0.7384	0.9544	0.5763	0.4896	0.9818	0.9593	0.8244
	3	0.9875	0.8624	0.8290	0.9956	0.6204	0.5516	0.9909	0.9912	0.9377

Table 13

Estimated power of Um1 for normal distribution.

q or 1−q		0.1			0.3			0.5
	m
n1, n2	θ	0	1	2	0	1	2	0	1	2
	1.5	0.2213	0.2109	0.2016	0.1710	0.1409	0.1280	0.1838	0.2523	0.2137
10, 10	2	0.3830	0.3613	0.3329	0.3034	0.2825	0.2513	0.3186	0.4001	0.3662
	2.5	0.4910	0.4805	0.4645	0.3878	0.3619	0.3127	0.4216	0.5127	0.4623
	3	0.5405	0.5318	0.5298	0.4481	0.4132	0.3809	0.4980	0.5778	0.5319
	1.5	0.3166	0.3095	0.2949	0.1850	0.1644	0.1425	0.1965	0.2712	0.2351
15, 10	2	0.3902	0.3720	0.3590	0.3114	0.3003	0.2808	0.3206	0.4224	0.3683
	2.5	0.5178	0.5103	0.5017	0.4067	0.3825	0.3413	0.4308	0.5334	0.4877
	3	0.5613	0.5527	0.5493	0.4660	0.4335	0.4115	0.5336	0.5966	0.5622
	1.5	0.3280	0.3305	0.3160	0.1943	0.1808	0.1640	0.2095	0.2879	0.2523
15, 15	2	0.4019	0.3909	0.3825	0.3642	03224	0.3025	0.3914	0.4598	0.4312
	2.5	0.5421	0.5358	0.5272	0.4865	0.4236	0.3738	0.5287	0.5971	0.5534
	3	0.5744	0.5819	0.5935	0.5112	0.4748	0.4526	0.6509	0.6603	0.6578
	1.5	0.3401	0.3366	0.3271	0.2098	0.2019	0.1839	0.2225	0.2997	0.2660
20, 15	2	0.4429	0.4221	0.4066	0.4013	0.3604	0.3246	0.4124	0.4772	0.4455
	2.5	0.5765	0.5639	0.5433	0.5222	0.4757	0.4215	0.5524	0.6214	0.5815
	3	0.6779	0.6525	0.6240	0.5620	0.5354	0.4918	0.6812	0.7009	0.6924
	1.5	0.3728	0.3650	0.3352	0.2201	0.2160	0.2007	0.2344	0.3116	0.2778
20, 20	2	0.4813	0.4635	0.4423	0.4247	0.3993	0.3743	0.4345	0.5972	0.4819
	2.5	0.6116	0.6053	0.5819	0.6057	0.5388	0.4710	0.6078	0.6813	0.6427
	3	0.7105	0.6996	0.6626	0.6108	0.5813	0.5520	0.7345	0.7724	0.7535
	1.5	0.3869	0.3744	0.3625	0.2377	0.2279	0.2215	0.2526	0.3319	0.2893
25, 20	2	0.5292	0.5037	0.4896	0.4589	0.4334	0.4017	0.4775	0.5588	0.5110
	2.5	0.6402	0.6206	0.6035	0.6260	0.5893	0.5280	0.6593	0.7179	0.6823
	3	0.7512	0.7300	0.6947	0.6613	0.6224	0.5903	0.7842	0.8308	0.8009
	1.5	0.4683	0.4635	0.4580	0.2565	0.2412	0.2328	0.2743	0.3524	0.3120
25, 25	2	0.5810	0.5222	0.5099	0.4991	0.4624	0.4456	0.5148	0.5977	0.5546
	2.5	0.6811	0.6690	0.6413	0.6778	0.6383	0.5779	0.7011	0.7685	0.7407
	3	0.8203	0.7991	0.7340	0.7124	0.6735	0.6329	0.8222	0.8533	0.8335
	1.5	0.4891	0.4828	0.4799	0.2756	0.2589	0.2501	0.2875	0.3713	0.3490
30, 30	2	0.6224	0.6015	0.5876	0.5463	0.5013	0.4897	0.5691	0.6364	0.6012
	2.5	0.7320	0.7056	0.6737	0.7318	0.6879	0.6388	0.7693	0.7922	0.7806
	3	0.8913	0.8843	0.8715	0.7639	0.7320	0.6879	0.8773	0.8997	0.8812
	1.5	0.5120	0.5007	0.4914	0.3144	0.2892	0.2785	0.3670	0.4009	0.3850
40, 40	2	0.6736	0.6507	0.6219	0.6017	0.5625	0.5382	0.6852	0.7355	0.7180
	2.5	0.7762	0.7519	0.7395	0.7923	0.7634	0.7179	0.8048	0.8125	0.8005
	3	0.9427	0.9266	0.9146	0.8514	0.8017	0.7685	0.9530	0.9770	0.9640

Table 14

Estimated power of Um1 for Cauchy distribution.

q or 1−q		0.1			0.3			0.5
	m
n1, n2	θ	0	1	2	0	1	2	0	1	2
	1.5	0.2819	0.2357	0.2015	0.3054	0.2142	0.1517	0.2597	0.2117	0.1621
10, 10	2	0.3912	0.3419	0.3220	0.4206	0.3230	0.2967	0.3723	0.3229	0.3016
	2.5	0.5519	0.4910	0.4517	0.5736	0.4725	0.4114	0.5068	0.4622	0.4399
	3	0.6306	0.6070	0.5830	0.6519	0.5812	0.5520	0.6210	0.4892	0.4624
	1.5	0.3010	0.2614	0.2487	0.3500	0.2443	0.1943	0.2793	0.2225	0.2219
15, 10	2	0.4349	0.3767	0.3543	0.4517	0.3611	0.3272	0.3992	0.3616	0.3590
	2.5	0.5720	0.5120	0.4871	0.5998	0.5019	0.4612	0.5364	0.5010	0.4711
	3	0.6910	0.6385	0.6220	0.6909	0.6057	0.5711	0.6619	0.2399	0.5168
	1.5	0.3450	0.2875	0.2676	0.3809	0.2698	0.2319	0.2941	0.2540	0.2410
15, 15	2	0.4629	0.4250	0.4006	0.4928	0.4095	0.3630	0.4431	0.4210	0.3803
	2.5	0.5810	0.5577	0.5230	0.6347	0.5367	0.4915	0.5679	0.5334	0.5116
	3	0.7565	0.6611	0.6498	0.7325	0.6348	0.6028	0.7320	0.5688	0.5499
	1.5	0.3629	0.3117	0.2989	0.4110	0.3001	0.2610	0.3389	0.2913	0.2813
20, 15	2	0.4930	0.4478	0.4223	0.5333	0.4373	0.3979	0.4818	0.4529	0.4426
	2.5	0.6044	0.5883	0.5622	0.6759	0.5771	0.5351	0.5990	0.5637	0.5437
	3	0.7977	0.6850	0.6733	0.7710	0.6692	0.6377	0.7997	0.5986	0.5818
	1.5	0.3920	0.3408	0.3258	0.4419	0.3298	0.2912	0.3614	0.3277	0.3180
20, 20	2	0.5450	0.4826	0.4647	0.5740	0.4636	0.4320	0.5227	0.4915	0.4712
	2.5	0.6570	0.6219	0.6008	0.7123	0.6029	0.5678	0.6390	0.6123	0.6013
	3	0.8419	0.7184	0.7029	0.8229	0.6980	0.6636	0.8334	0.6378	0.6207
	1.5	0.4347	0.3775	0.3660	0.4722	0.3412	0.3161	0.4010	0.3610	0.3420
25, 20	2	0.5661	0.5210	0.5110	0.5934	0.5117	0.4707	0.5666	0.5409	0.5218
	2.5	0.6955	0.6506	0.6336	0.7436	0.6320	0.5830	0.6813	0.6618	0.6407
	3	0.8910	0.7519	0.7375	0.8651	0.7321	0.6923	0.8740	0.7024	0.6515
	1.5	0.4891	0.4256	0.4095	0.4931	0.4002	0.3781	0.4398	0.4018	0.3810
25, 25	2	0.5997	0.5412	0.5278	0.6247	0.5290	0.5079	0.6050	0.5799	0.5658
	2.5	0.7338	0.6878	0.6550	0.7725	0.6627	0.6185	0.7208	0.7037	0.6997
	3	0.9316	0.7818	0.7655	0.9021	0.7617	0.7259	0.9089	0.7128	0.7011
	1.5	0.5378	0.4610	0.4503	0.5240	0.4495	0.4179	0.4872	0.4634	0.4410
30, 30	2	0.6412	0.5899	0.5619	0.6755	0.5776	0.5446	0.6613	0.6260	0.6160
	2.5	0.7745	0.7398	0.7147	0.7991	0.7194	0.6756	0.7719	0.7585	0.7315
	3	0.9512	0.8591	0.8329	0.9503	0.8360	0.7607	0.9385	0.7687	0.7585
	1.5	0.5660	0.5090	0.4887	0.5538	0.4979	0.4638	0.5214	0.4966	0.4891
40, 40	2	0.6823	0.6413	0.6257	0.7135	0.6320	0.5824	0.7229	0.6897	0.6690
	2.5	0.8421	0.7829	0.7592	0.8356	0.7728	0.7188	0.8320	0.8119	0.8094
	3	0.9789	0.9127	0.8856	0.9727	0.9005	0.8222	0.9649	0.8430	0.8390

Table 15

Estimated power of Um2 for uniform distribution.

q or 1−q		0.1			0.3			0.5
	m
n1, n2	θ	0	1	2	0	1	2	0	1	2
	1.5	0.4218	0.2824	0.2767	0.2993	0.2615	0.1827	0.2776	0.2118	0.2113
10, 10	2	0.6494	0.5105	0.5054	0.5189	0.4922	0.3213	0.5052	0.3051	0.2808
	2.5	0.6986	0.6779	0.6625	0.6802	0.6431	0.4385	0.6615	0.3867	0.3016
	3	0.8115	0.7865	0.7750	0.7976	0.7665	0.5211	0.7764	0.4777	0.3265
	1.5	0.4523	0.2878	0.2845	0.3224	0.2817	0.2135	0.2908	0.2420	0.2217
15, 10	2	0.6611	0.5451	0.5312	0.5537	0.5209	0.3623	0.5381	0.3446	0.2907
	2.5	0.7452	0.7277	0.7115	0.7392	0.7013	0.4612	0.7182	0.4175	0.3285
	3	0.8562	0.8268	0.8121	0.8302	0.8024	0.5448	0.8236	0.4760	0.4218
	1.5	0.4817	0.3248	0.3156	0.3536	0.3101	0.2409	0.3332	0.2634	0.2368
15, 15	2	0.6903	0.6413	0.6370	0.6495	0.6205	0.3925	0.6386	0.3752	0.3119
	2.5	0.8254	0.8104	0.7923	0.8122	0.7724	0.5102	0.8095	0.4523	0.3420
	3	0.9193	0.9054	0.8828	0.9097	0.8619	0.5664	0.9005	0.5277	0.7525
	1.5	0.5229	0.3660	0.3604	0.3813	0.3551	0.2652	0.3772	0.2850	0.2599
20, 15	2	0.7217	0.6899	0.6771	0.6982	0.6610	0.4178	0.6828	0.4468	0.3425
	2.5	0.8728	0.8587	0.8392	0.8617	0.8235	0.5389	0.8538	0.5635	0.3934
	3	0.9457	0.9305	0.9197	0.9365	0.9029	0.6149	0.9274	0.6564	0.5537
	1.5	0.5516	0.3852	0.3798	0.4106	0.3744	0.2911	0.3945	0.2945	0.2798
20, 20	2	0.7543	0.7403	0.7292	0.7499	0.7120	0.4325	0.7353	0.4779	0.3937
	2.5	0.9102	0.8992	0.8714	0.9010	0.8536	0.5703	0.8957	0.6140	0.4741
	3	0.9713	0.9648	0.9579	0.9699	0.9411	0.6577	0.9604	0.7175	0.6210
	1.5	0.5920	0.4476	0.4261	0.4495	0.4058	0.3266	0.4449	0.3257	0.3044
25, 20	2	0.7997	0.7837	0.7699	0.7981	0.7539	0.4698	0.7762	0.5450	0.4528
	2.5	0.9356	0.9297	0.8915	0.9344	0.8748	0.6027	0.9253	0.7119	0.5433
	3	0.9855	0.9799	0.9757	0.9801	0.9625	0.6903	0.9736	0.7984	0.7068
	1.5	0.6317	0.4892	0.4644	0.4927	0.4436	0.3702	0.4837	0.3346	0.3156
25, 25	2	0.8322	0.8275	0.8113	0.8306	0.7976	0.4984	0.8129	0.5703	0.4917
	2.5	0.9614	0.9501	0.9280	0.9578	0.9017	0.6378	0.9496	0.7427	0.6334
	3	0.9927	0.9884	0.9808	0.9906	0.9701	0.7286	0.9836	0.8569	0.7705
	1.5	0.7166	0.5880	0.5522	0.5995	0.5216	0.3916	0.5613	0.3760	0.3567
30, 30	2	0.8899	0.8799	0.8684	0.8817	0.8513	0.5399	0.8778	0.6412	0.5672
	2.5	0.9894	0.9796	0.9640	0.9836	0.9422	0.6778	0.9747	0.8039	0.6945
	3	0.9959	0.9943	0.9895	0.9950	0.9811	0.7790	0.9932	0.8920	0.8217
	1.5	0.8002	0.6203	0.5978	0.6324	0.5727	0.4328	0.6029	0.5078	0.4184
40, 40	2	0.9478	0.9224	0.9109	0.9335	0.8997	0.5700	0.9134	0.8133	0.6585
	2.5	0.9965	0.9902	0.9809	0.9918	0.9678	0.7102	0.9890	0.9547	0.8279
	3	0.9987	0.9974	0.9920	0.9979	0.9899	0.8588	0.9967	0.9922	0.9391

Table 16

Estimated power of Um2 for normal distribution.

q or 1−q		0.1			0.3			0.5
	m
n1, n2	θ	0	1	2	0	1	2	0	1	2
	1.5	0.2299	0.2528	0.2814	0.2317	0.2270	0.2178	0.1790	0.2517	0.2111
10, 10	2	0.3907	0.3810	0.4077	0.3644	0.3583	0.3497	0.3180	0.4025	0.3609
	2.5	0.5150	0.5212	0.5386	0.4780	0.4709	0.4510	0.4207	0.5113	0.4665
	3	0.5590	0.5824	0.6243	0.5613	0.5566	0.5414	0.5337	0.5709	0.5370
	1.5	0.3192	0.3214	0.3441	0.2677	0.2578	0.2440	0.1999	0.2765	0.2396
15, 10	2	0.3983	0.4131	0.4319	0.3912	0.3822	0.3787	0.3122	0.4202	0.3693
	2.5	0.5252	0.5336	0.5613	0.4944	0.4913	0.4877	0.4340	0.5398	0.4895
	3	0.6080	0.6329	0.6837	0.6119	0.6002	0.5944	0.5320	0.5910	0.5615
	1.5	0.3356	0.3417	0.3619	0.2998	0.2907	0.2668	0.2060	0.2893	0.2517
15, 15	2	0.4443	0.4734	0.5013	0.4502	0.4376	0.4219	0.3918	0.4528	0.4399
	2.5	0.5492	0.5637	0.6027	0.5413	0.5317	0.5299	0.5214	0.5987	0.5596
	3	0.6891	0.7252	0.7614	0.6927	0.6829	0.6752	0.6574	0.6616	0.6582
	1.5	0.3494	0.3523	0.3952	0.3008	0.2997	0.2883	0.2241	0.2952	0.2626
20, 15	2	0.4609	0.4912	0.5387	0.4767	0.4599	0.4228	0.4117	0.4713	0.4495
	2.5	0.6360	0.6514	0.6926	0.6372	0.6302	0.6212	0.5504	0.6291	0.5895
	3	0.7396	0.7790	0.8052	0.7416	0.7310	0.7243	0.6890	0.7025	0.6993
	1.5	0.3602	0.3915	0.4337	0.3225	0.3185	0.3004	0.2311	0.3775	0.2701
20, 20	2	0.5389	0.5667	0.6012	0.5465	0.5311	0.5110	0.4364	0.5919	0.4809
	2.5	0.6592	0.6921	0.7331	0.6610	0.6550	0.6494	0.6081	0.6808	0.6489
	3	0.7884	0.8395	0.8544	0.7917	0.7805	0.7710	0.7367	0.7779	0.7513
	1.5	0.3898	0.4309	0.4702	0.3652	0.3541	0.3319	0.2533	0.3384	0.2880
25, 20	2	0.5686	0.5914	0.6508	0.5713	0.5660	0.5330	0.4847	0.5556	0.5120
	2.5	0.6991	0.7316	0.7802	0.7053	0.6909	0.6895	0.6625	0.7134	0.6895
	3	0.8497	0.8825	0.9009	0.8541	0.8444	0.8319	0.7877	0.8338	0.8094
	1.5	0.4705	0.4777	0.4993	0.3962	0.3819	0.3540	0.2789	0.3579	0.3116
25, 25	2	0.5912	0.6235	0.6626	0.6021	0.5891	0.5680	0.5162	0.5908	0.5525
	2.5	0.7399	0.7629	0.7941	0.7413	0.7310	0.7284	0.7088	0.7610	0.7430
	3	0.8785	0.9014	0.9325	0.8820	0.8711	0.8614	0.8240	0.8519	0.8398
	1.5	0.4908	0.4991	0.5395	0.4347	0.4210	0.3917	0.2821	0.3766	0.3411
30, 30	2	0.6289	0.6314	0.6997	0.6118	0.6009	0.5883	0.5640	0.6365	0.6014
	2.5	0.7695	0.7920	0.8448	0.7729	0.7618	0.7591	0.7699	0.7927	0.7890
	3	0.9056	0.9310	0.9499	0.9117	0.9190	0.9020	0.8792	0.8997	0.8899
	1.5	0.5258	0.5440	0.5865	0.4698	0.4608	0.4319	0.3617	0.4075	0.3868
40, 40	2	0.6783	0.6939	0.7290	0.6329	0.6197	0.6108	0.6878	0.7393	0.7186
	2.5	0.8220	0.8580	0.8985	0.8360	0.8115	0.8090	0.8065	0.8157	0.8016
	3	0.9569	0.9735	0.9798	0.9688	0.9526	0.9508	0.9538	0.9771	0.9645

Table 17

Estimated power of Um2 for Cauchy distribution.

Based on the power computations, we have the following observations:

For uniform distribution, change in scale of the order of 3 is detected:
1. For q=0.5, for random samples of size ≥40 for Um1 and Um2 tests at m=0.
2. For q≠0.5, change in scale of same order is detected for random samples of size ≥30 for Um2 tests at m=0. This authenticates the observation 1, of Section 4.
For normal distribution, change in scale of the order of 3 is detected:
1. For q∈0.3, 0.7, for random samples of size ≥20 for Um1 and Um2 tests at m=0
2. For q∉0.3, 0.7, the change in scale of same order is detected for random samples of size ≥20 for Um2 tests at m=0, 1, 2 with maximum power is achieved at m=0. This authenticates the observation 2, of Section 4.
For Cauchy distribution, change in scale of the order of 3 is detected:
1. For q=0.5, for random samples of size ≥40 for Um1 and Um2 tests at m=0, 1, 2 with maximum power is achieved at m=1.
2. For q≠0.5, the change in scale of same order is detected at random samples of size ≥40, for Um2 tests at m = 0, 1, 2 with maximum power is achieved at m=2 for q∉0.3, 0.7, otherwise maximum power is achieved at m=0. This authenticates the observation 3, of Section 4.
Also the power of Um2 test is equivalent to Um1 test for q=0.5, and the power of Um2 test is greater than Um1 test for q≠0.5, for all choices of m. This authenticates the observation 4, of Section 4.
For all other Um1 and Um2 tests, one needs to take larger sample size, to detect the change of scale of the same order. This once again authenticates the computations of AREs as well.

ACKNOWLEDGMENTS

The authors thank the editor-in-chief and anonymous referee for their valuable suggestions, which led to an improvement of the paper.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 2
Pages: 155 - 170
Publication Date: 2019/06/04
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TY  - JOUR
AU  - Manish Goyal
AU  - Narinder Kumar
PY  - 2019
DA  - 2019/06/04
TI  - A Generalization of the Sukhatme's Test for Two-Sample Scale Problem
JO  - Journal of Statistical Theory and Applications
SP  - 155
EP  - 170
VL  - 18
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190524.002
DO  - 10.2991/jsta.d.190524.002
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