Journal of Statistical Theory and Applications

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.190524.002How 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, 0q1. 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:
HA : θ>1.

Under the null hypothesis, Xs and Ys are alike, but under the alternative, Ys will have more variation than Xs.

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 [810], 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  12m+1n1 and 1jn2. Define the following two kernels  h1 and h2 as

h1X1,,X2m+1;Yj=1,if0MXYj  and Xi,Yj0;i=1,,2m+1  orYjMX0 and Xi,Yj0;i=1,,2m+10,    otherwise
h2X1,,X2m+1;Yj=1,if0MXYj  and Xi,Yj0;i=1,,2m+1     or YjMX0 and Xi,Yj0;i=1,,2m+11,if0YjMX and Xi,Yj0;i=1,,2m+1  orMXYj0 and Xi,Yj0;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

Um(c)X1,,Xn1;Y1,,Yn2=n12m+1n211sj=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

  1. For m=0, the test statistics Um1 corresponds to test statistics of Sukhatme [2].

  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+1n211cj=1n2Eh(c)Xi1,,Xi2m+1;Yj=Eh(c)Xi1,,Xi2m+1;Yj.

For c=1,

EUm1=0PMXtPYj=tdt+0PMXtPYj=tdt.

Under H0,

EUm1=121q2m+2+q2m+2.

For c=2,

EUm2=0PMXtPYj=tdt+0PMXtPYj=tdt.0PMXtPYj=tdt0PMXtPYj=tdt.

Under H0,

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/Nλ,  0<λ<1, is normal with mean zero and variance, σ2Um(c), as

σ2Um(c)=2m+12ζ10(c)λ+ζ01(c)1λ.

Here

ζ10(c)=Eh(c)x,X2,,X2m+1;Y12EUm(c)2
and
ζ01(c)=Eh(c)X1,X2,,X2m+1;y2EUm(c),
where
h(c)x,X2,,X2m+1;Y1=Eh(c)X1,X2,X2m+1;Y1|X1=x
and
h(c)X1,X2,,X2m+1;y=Eh(c)X1,X2,X2m+1;Y1|Y1=y.

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

σ02Um(c)=2m+12ρm(c)λ1λ,
where for c=1,
ρm1=i=m+12mj=m+12m2mi2mjr=0is=0jirjs1i+j+r+s2mr+12ms+1+22mmi=m+12m2mir=0iir1i+r2mr+1×s=0mms1m+s2ms+12ms+2+2mm2r=0ms=0mmrms1r+s2mr+12ms+14mrs+3×1q4m+3+q4m+3EUm12
and for c=2,
ρm2=2mm24k=0ml=0mmkml1k+l2mk+12ml+14mkl+34l=0mml1m+l2ml+12ml+2×k=0mmk1m+k2mk+1+k=0ml=0mmkml1k+l2mk+12ml+1×1q4m+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=N1/2θ, is

e2Um(c)=limNddθEUm(c)|θ=12Nσ02Um(c)=c22m+12mm2σ02Um(c)0Fyqm1Fymf  2ydy0FymqFymf  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

e2S=λ1λ112q21q2|y|f  2ydy2e2DK=4λ1λ13q1q|y|f  2ydy2e2K=112λ1λq7+1q7|y|Fyq2f  2ydy2e2MGA=831600λ1λ131q11+1q110yF2yqFy2f  2ydy0yFyq21Fy2f  2ydy2e2Tk=4k24k1λ1λq4k1+1q4k1|y|Fyq2k2f  2ydy2.

In the following Tables 110, we have computed the AREs of Um1 and Um2 tests w.r.t. competing tests for some underlying distributions.

Test
q or 1q 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 1q 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 1q 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 1q 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 1q 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 1q 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 1q 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 1q 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 1q 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 1q 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:

  1. 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 m2 with maximum efficiency achieved at m=0.

    2. For q0.5, Um2 test is more efficient than S test for m=0 and is more efficient than MGA test for m2 with maximum efficiency achieved at m=0.

  2. 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 q0.3,0.7, then Um1 and Um2 tests are more efficient than S and MGA test for m=0.

    2. For q0.3,0.7, Um2 test is more efficient than S and MGA tests for m2, with maximum efficiency achieved at m=0.

  3. 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 m2 with maximum efficiency achieved at m=1.

    2. For q0.5, Um2 tests are more efficient than its competing tests with maximum efficiency is for m as large as possible for q0.3,0.7, otherwise the maximum efficiency for Um2 tests is achieved at m=0.

  4. 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 q0.5. 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, Um1m=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=1.50.53 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 1217.

q or 1q 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 1q 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 1q 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 1q 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 1q 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 1q 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:

  1. 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 q0.5, 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.

  2. For normal distribution, change in scale of the order of 3 is detected:

    1. For q0.3,0.7, for random samples of size 20 for Um1 and Um2 tests at m=0

    2. For q0.3,0.7, 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.

  3. 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 q0.5, 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 q0.3,0.7, otherwise maximum power is achieved at m=0. This authenticates the observation 3, of Section 4.

  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 q0.5, for all choices of m. This authenticates the observation 4, of Section 4.

  5. 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.

REFERENCES

8.R. Tamura, Bull. Math. Stat., Vol. 9, 1960, pp. 61-67. http://hdl.handle.net/2324/12995
9.R. Tamura, Bull. Math. Stat., Vol. 10, 1962, pp. 31-38. http://hdl.handle.net/2324/13005
10.R. Tamura, Bull. Math. Stat., Vol. 12, 1966, pp. 89-94. http://hdl.handle.net/2324/13020
11.T. Yanagawa, Bull. Math. Stat., Vol. 14, 1970, pp. 15-24. http://hdl.handle.net/2324/13041
18.K. Kusum, J. Indian Stat. Assoc., Vol. 23, 1985, pp. 97-107.
19.K.L. Mehra and K.S. Rao, Goa University, in Proceeding of 80th Indian Science Congress (Goa), 1992.
22.D.J. Hand, F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski, A Handbook of Small Datasets, Chapman & Hall, London, 1994. https://www.crcpress.com/A-Handbook-of-Small-Data-Sets/Hand-Daly-McConway-Lunn-Ostrowski/p/book/9780412399206
Journal
Journal of Statistical Theory and Applications
Volume-Issue
18 - 2
Pages
155 - 170
Publication Date
2019/06/04
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.190524.002How to use a DOI?
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/).

Cite this article

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
ID  - Goyal2019
ER  -