Journal of Statistical Theory and Applications

Volume 17, Issue 4, December 2018, Pages 647 - 660

A Comprehensive Study on Power of Tests for Normality

Authors
Hadi Alizadeh Noughabi1, *
1Department of Statistics, University of Birjand, Birjand, Iran
*

Corresponding author. Email: alizadehhadi@birjand.ac.ir

Received 14 October 2017, Accepted 17 May 2018, Available Online 31 December 2018.
DOI
10.2991/jsta.2018.17.4.7How to use a DOI?
Keywords
Test of normality; Monte Carlo simulation; Power of test; The generalized lambda distribution
Abstract

Many statistical procedures assume that the underling distribution is normal. In this paper, we consider the popular and powerful tests for normality and investigate the power values of these tests to detect deviations from normality. The family of four-parameter generalized lambda distributions (FMKL) for its high flexibility is considered as alternative distributions. We then compare the power values of normality tests against these alternatives and for different sample sizes. The considered tests are Kolmogorov-Smirnov, Anderson-Darling, Kuiper, Jarque-Bera, Cramer von Mises, Shapiro-Wilk and Vasicek. These tests are popular tests which are commonly used in practice and statistical software. The tests are described and then power values of the tests are compared against FMKL family by Monte Carlo simulation. The results are discussed and interpreted. Finally, we apply some real data examples to show the behavior of the tests in practice.

Copyright
© 2018 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. INTRODUCTION

Ramberg and Schmeiser [1] introduced the four-parameter generalized lambda distribution (GLD) as

Q(u)=λ1+uλ3(1u)λ4λ2,
where Q(u) is the quantile function, 0u, λ1, λ2 are the location and scale parameters, and λ3, λ4 are shape parameters jointly related to the strengths of the lower and upper tails, respectively. For its high flexibility it is used in many fields such as modeling financial data.

Because of the limitations of the Ramberg and Schmeiser (RS) parameterizations, Freimer et al. [2] proposed a new parameterization called FMKL as

Q(u)=λ1+1λ2uλ31λ3(1u)λ41λ4,
where 0u1, λ1, λ2 are the location and scale parameters, respectively. Also λ3 and λ4 determine the shape characteristics and for a symmetric distribution λ3=λ4.

The five different shapes of the FMKL are: unimodal, U-shaped, J-shaped, S-shaped, and monotone, which may be symmetric and asymmetric with smooth, abrupt, truncated, long, medium or short tails.

In many situations, a goodness of fit test about the distribution of the population using observations is necessary. Since the normal distribution is widely used in many statistical procedures and also is the most fundamental distribution, test for the normal hypothesis is indispensable. Moreover, testing normality is one of the most areas of statistical research. For example in statistical modeling the normal assumption of the underlying error distribution must be checked. Therefore, many tests for normality are proposed by authors. A fair of normality tests can be found in the statistical literature. In many situations, a goodness of fit test about the distribution of the population is necessary. Since the normal distribution is the most basic distribution and use widely in statistics, test for the normal hypothesis has been studied by many authors. See for example, D'Agostino and Stephens [3], Huber-Carol et al. [4], Thode [5].

Recently, testing normality has been considered by Alizadeh Noughabi and Arghami [6], Harri and Coble [7], Sanqui et al. [8], Zamanzade and Arghami [9], Marmolejo-Ramos and González-Burgos [10], Joenssen and Vogel [11] and Wang [12].

In this article, we consider seven popular (like Kolmogorov-Smirnov, Anderson-Darling) and powerfulness (like Shapiro-Wilk, Vasicek) normality tests and compare power values of these tests against the GLD (FMKL) with different parameters. We show that no single test procedure is uniformly more powerful than others. However, the powerful tests can be determined based on type of alternatives. Thus, tests for normality based on type of alternatives are classified.

The methodologies of the considered tests are presented in Section 2. Power values of the tests are compared with each other against the FMKL family by Monte Carlo simulation in Section 3. In Section 4, the applicability of the tests in practice is shown by real data. Finally, some conclusions are given in Section 5.

2. TESTS FOR NORMALITY

Given a random sample X1,,Xn from a continuous probability distribution F with a density f(x), over the real line and with mean μ and variance σ2<, the hypothesis of interest is

H0:f(x)=f0(x;μ,σ)=12πσexp12xμσ2,  for some (μ,σ)Θ,x
where μ and σ are unspecified and Θ=×+. The alternative to H0 is
H1:f(x)f0(x;μ,σ)  for any (μ,σ)Θ.

In this section, we consider seven popular tests for the above hypothesis. The considered tests are Cramer von Mises [13], Kolmogorov-Smirnov [14], Anderson-Darling [15], Kuiper [16], Shapiro-Wilk [17], Vasicek [18] and Jarque-Bera [19]. These tests are commonly used in practice and software. For example, the Shapiro-Wilks test is used in SAS software for testing normality. The description of each normality tests is presented in Table 1.

Test of normality Test statistic Notations
1- Cramer von Mises CH=112n+i=1n2i12nZi2;i=1,,n. Zi=ΦX(i)X¯SX: where Φ is the cdf of standard normal distribution.
2- Kolmogorov-Smirnov D+=maxinZi,D=maxZii1n,i=1,,nD=max(D+,D) Zi=ΦX(i)X¯SX: where Φ is the cdf of standard normal distribution.
3- Kuiper V=D++D D+ and D are as above.
4- Shapiro-Wilk W=i=1n/2a(ni+1)(X(ni+1)X(i))2i=1n(X(i)X¯)2 The coefficients ai are tabulated in Pearson and Hartley [20]
5- Anderson-Darling A2=ni=1n(2i1)ln(Zi)+ln(1Zni+1)n Zi=ΦX(i)X¯SX: where Φ is the cdf of standard normal distribution.
6- Vasicek KLmn=expH(n,m)SXH(n,m)=1ni=1nlogn2m(X(i+m)X(im)) SX2: sample variance m: positive integer mn2
7- Jarque-Bera JB=nc26+(k3)224 c=skewnessk=kurtosis
Table 1

Tests of normality.

From the aforementioned tests, Vasicek's test, Shapiro-Wilk and Jarque-Bera test are specific in the sense that the null hypothesis is normal, while the rest are suitable for any null family of distributions. For further study about this tests, see D'Agostino and Stephens [3] and references there in.

3. SIMULATION STUDY

In this section, Type I error of the tests are obtained and then power values of the tests against flexible FMKL family are computed through a Monte Carlo simulation.

3.1 Type I Error of the Tests

Through a Monte Carlo simulation, we compute Type I error of the tests. Table 2 presents Type I error probabilities (the actual size of the tests), which have been obtained by 20,000 simulations. As is evident from Table 2 the actual size of the tests are quite acceptable.

Test of normality σ = .01 σ = .25 σ = 1 σ = 5 σ = 10 σ = 100
Cramer von Mises 0.0510 0.0506 0.0498 0.0492 0.0485 0.0509
Kolmogorov-Smirnov 0.0507 0.0492 0.0503 0.0505 0.0491 0.0511
Kuiper 0.0517 0.0509 0.0501 0.0493 0.0508 0.0487
Shapiro-Wilk 0.0503 0.0497 0.0499 0.0504 0.0497 0.0501
Anderson-Darling 0.0491 0.0505 0.0497 0.0506 0.0490 0.0508
Vasicek 0.0498 0.0501 0.0503 0.0499 0.0502 0.0496
Jarque-Bera 0.0503 0.0495 0.0501 0.0499 0.0502 0.0501
Table 2

Actual sizes of the tests for nominal significance level α = 0.05 (n = 20, σ = standard deviation).

3.2 Shapes of FMKL Family

The shapes returned by FMKL family are classified by Freimer et al. [2] as follows.

  • Class I λ3<1,λ4<1: Unimodal densities with continuous tails. This class can be subdivided with respect to the finite or infinite slopes of the densities at the end points.

    • Class Ia λ3,λ40.5,

    • Class Ib 0.5<λ3<1,λ40.5,

    • Class Ic 0.5<λ3<1,0.5<λ4<1.

  • Class II λ3>1,λ4<1: Monotone pdfs similar to those of the exponential or Chi-square distributions. The left tail is truncated.

  • Class III 1<λ3<2,1<λ4<2: U-shaped densities with both tails truncated.

  • Class IV λ3>2,1<λ4<2: Rarely occurring S-shaped pdfs with one mode and one antimode. Both tails are truncated.

  • Class V λ3>2,λ4>2: Unimodal pdfs with both tails truncated.

Examples of each class of shapes are displayed in Figs. 17.

Figure 1

Class Ia pdfs including the normal distribution.

Figure 2

Class Ib pdfs.

Figure 3

Class Ic pdfs.

Figure 4

Class II pdfs includes the exponential distribution.

Figure 5

Class III U-shaped pdfs.

Figure 6

Class IV S-shaped pdfs.

Figure 7

Class V pdfs.

3.3 Power Comparison

To comparison of the power values of the tests, we select alternatives from FMKL family with different parameters. We compute the power values of the tests based on CH, D, V, W, A2, KLmn and JB statistics against FMKL family by means of Monte Carlo simulations. As mentioned above, the alternatives can divide into five groups (Class I to V).

Power values of the tests were obtained by simulation in the following manner.

Under each alternative we generated 20,000 samples of size 10, 20, 30, 50. We calculated for each sample the statistics (CH, D, V, W, A2, KLmn, JB) and the power value of the corresponding test was computed by the frequency of the event “the value of the test statistic is in the critical region”. The required critical regions are given in the corresponding articles, but we obtained them by simulation, before power simulations. The power values of the tests are presented in Tables 39. For each sample size and alternative, the bold type in these Tables indicates the test achieving the maximal power. For Vasicek's test, we used the recommended window sizes proposed by Vasicek [18], i.e., m = 2, (3), 4 for sample sizes n = 10, (20, 30), 50, respectively.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 −0.2 −0.2 0 22.21 0.1528 0.1549 0.1735 0.1780 0.0866 0.1771 0.2036
0.25 0.25 0 2.54 0.0457 0.0463 0.0461 0.0441 0.0558 0.0427 0.0331
0.5 0.5 0 2.08 0.0464 0.0479 0.0452 0.0459 0.0876 0.0454 0.0227
−0.2 −0.1 2.63 35.40 0.1313 0.1315 0.1467 0.1512 0.0787 0.1507 0.1736
−0.2 0 4.65 73.8 0.1278 0.1280 0.1419 0.1474 0.0843 0.1519 0.1743
−0.2 0.25 6.99 157.98 0.1546 0.1561 0.1792 0.1879 0.1309 0.1994 0.2097
−0.2 0.5 7.25 189.98 0.2075 0.2184 0.2480 0.2616 0.2083 0.2811 0.2780
0 −0.1 −2.81 17.83 0.0931 0.0938 0.1017 0.1057 0.0653 0.1097 0.1277
0 0.25 1.05 3.70 0.0765 0.0741 0.0830 0.0872 0.664 0.0907 0.0944
0 0.5 0.566 2.40 0.1063 0.1092 0.1263 0.1340 0.1116 0.1438 0.1425
0.25 −0.1 −2.13 11.21 0.1071 0.1043 0.1226 0.1292 0.0888 0.1360 0.1478
0.25 0.5 −0.041 2.25 0.0509 0.0538 0.0540 0.0530 0.0769 0.0533 0.0363
0.5 −0.1 −1.15 4.09 0.1513 0.1562 0.1777 0.1908 0.1486 0.2052 0.2050
20 −0.2 −0.2 0 22.21 0.2405 0.2575 0.2906 0.3083 0.1376 0.3224 0.3672
0.25 0.25 0 2.54 0.0464 0.0488 0.0447 0.0419 0.0668 0.0377 0.0186
0.5 0.5 0 2.08 0.0501 0.0692 0.0612 0.0657 0.1628 0.0649 0.0064
−0.2 −0.1 2.63 35.40 0.1955 0.2089 0.2391 0.2543 0.1131 0.2657 0.3110
−0.2 0 4.65 73.8 0.1914 0.1970 0.2303 0.2485 0.1281 0.2684 0.3047
−0.2 0.25 6.99 157.98 0.2755 0.2729 0.3330 0.3613 0.2637 0.4003 0.4123
−0.2 0.5 7.25 189.98 0.3790 0.4069 0.4731 0.5137 0.4726 0.5747 0.5334
0 −0.1 −2.81 17.83 0.1195 0.1257 0.1436 0.1566 0.0712 0.1747 0.2127
0 0.25 1.05 3.70 0.0981 0.0925 0.1119 0.1231 0.0908 0.1481 0.1627
0 0.5 0.566 2.40 0.1741 0.1744 0.2234 0.2496 0.2360 0.3039 0.2719
0.25 −0.1 −2.13 11.21 0.1738 0.1688 0.2091 0.2309 0.1564 0.2661 0.2822
0.25 0.5 −0.041 2.25 0.0664 0.0686 0.0738 0.0773 0.1322 0.0807 0.0375
0.5 −0.1 −1.15 4.09 0.2680 0.2819 0.3443 0.3824 0.3481 0.4431 0.4049
30 −0.2 −0.2 0 22.21 0.3199 0.3535 0.3903 0.4132 0.2093 0.4289 0.4805
0.25 0.25 0 2.54 0.0449 0.0502 0.0480 0.0444 0.0741 0.0365 0.0105
0.5 0.5 0 2.08 0.0653 0.0937 0.0849 0.0934 0.2309 0.0986 0.0028
−0.2 −0.1 2.63 35.40 0.2695 0.2970 0.3300 0.3527 0.1789 0.3711 0.4263
−0.2 0 4.65 73.8 0.2668 0.2774 0.3254 0.3470 0.1962 0.3763 0.4219
−0.2 0.25 6.99 157.98 0.3847 0.3763 0.4599 0.4975 0.3857 0.5528 0.5561
−0.2 0.5 7.25 189.98 0.5343 0.5667 0.6508 0.6999 0.6765 0.7739 0.7052
0 −0.1 −2.81 17.83 0.1556 0.1656 0.1919 0.2087 0.1006 0.2288 0.2744
0 0.25 1.05 3.70 0.1292 0.1153 0.1526 0.1687 0.1196 0.2096 0.2200
0 0.5 0.566 2.40 0.2589 0.2591 0.3383 0.3878 0.3729 0.4775 0.3946
0.25 −0.1 −2.13 11.21 0.2399 0.2235 0.2933 0.3236 0.2312 0.3816 0.3894
0.25 0.5 −0.041 2.25 0.0867 0.0862 0.0996 0.1076 0.1776 0.1133 0.0354
0.5 −0.1 −1.15 4.09 0.3914 0.4127 0.5011 0.5576 0.5337 0.6466 0.5699
50 −0.2 −0.2 0 22.21 0.4480 0.5077 0.5451 0.5780 0.3270 0.6084 0.6562
0.25 0.25 0 2.54 0.0497 0.0552 0.0527 0.0523 0.0918 0.0437 0.0068
0.5 0.5 0 2.08 0.0966 0.1521 0.1396 0.1691 0.4101 0.2094 0.0095
−0.2 −0.1 2.63 35.40 0.3738 0.4194 0.4624 0.4946 0.2648 0.5398 0.5927
−0.2 0 4.65 73.8 0.3797 0.3957 0.4563 0.4879 0.2910 0.5347 0.5831
−0.2 0.25 6.99 157.98 0.5622 0.5449 0.6537 0.7002 0.5894 0.7737 0.7631
−0.2 0.5 7.25 189.98 0.7513 0.7961 0.8586 0.9030 0.9016 0.9484 0.9003
0 −0.1 −2.81 17.83 0.2081 0.2310 0.2669 0.2969 0.1355 0.3465 0.4017
0 0.25 1.05 3.70 0.1811 0.1511 0.2195 0.2494 0.1819 0.3327 0.3312
0 0.5 0.566 2.40 0.3920 0.3995 0.5127 0.5946 0.6172 0.7340 0.5962
0.25 −0.1 −2.13 11.21 0.3529 0.3282 0.4396 0.4870 0.3692 0.5861 0.5811
0.25 0.5 −0.041 2.25 0.1242 0.1237 0.1525 0.1777 0.2961 0.2181 0.0505
0.5 −0.1 −1.15 4.09 0.5807 0.6176 0.7171 0.7848 0.7895 0.8772 0.7932
Table 3

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group Ia.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 0.6 −0.2 −7.26 198.82 0.2239 0.2423 0.2734 0.2894 0.2451 0.3098 0.2951
0.6 −0.1 −0.94 3.29 0.1687 0.1799 0.2047 0.2190 0.1846 0.2398 0.2266
0.6 0 −0.42 2.16 0.1222 0.1292 0.1457 0.1556 0.1400 0.1704 0.1624
0.6 0.25 0.094 2.18 0.0595 0.0607 0.0639 0.0662 0.0942 0.0691 0.0492
0.6 0.5 0.045 2.03 0.0446 0.0515 0.0469 0.0486 0.0941 0.0479 0.0221
0.75 −0.2 −7.33 212.90 0.2540 0.2866 0.3140 0.3339 0.3070 0.3631 0.3360
0.75 −0.1 −0.69 2.60 0.1926 0.2121 0.2406 0.2584 0.2373 0.2809 0.2614
0.75 0 −0.24 1.95 0.1447 0.1587 0.1805 0.1925 0.1837 0.2120 0.1953
0.75 0.25 0.18 2.12 0.0757 0.0801 0.0833 0.0871 0.1201 0.0929 0.0681
0.75 0.5 0.12 1.99 0.0534 0.0605 0.0563 0.0583 0.1109 0.0579 0.0294
0.9 −0.2 −7.49 229.48 0.2764 0.3180 0.3471 0.3689 0.3569 0.4001 0.3577
0.9 −0.1 −0.49 2.20 0.2223 0.2554 0.2803 0.2993 0.2882 0.3257 0.2921
0.9 0 −0.09 1.84 0.1673 0.1906 0.2109 0.2278 0.2290 0.2482 0.2186
0.9 0.25 0.26 2.11 0.0913 0.0995 0.1047 0.1121 0.1460 0.1220 0.0869
0.9 0.5 0.19 1.97 0.0633 0.0747 0.0710 0.0739 0.1349 0.0780 0.0397
20 0.6 −0.2 −7.26 198.82 0.4219 0.4648 0.5269 0.5732 0.5592 0.6399 0.5715
0.6 −0.1 −0.94 3.29 0.3120 0.3412 0.4032 0.4495 0.4458 0.5179 0.4501
0.6 0 −0.42 2.16 0.2105 0.2228 0.2727 0.3105 0.3202 0.3750 0.3107
0.6 0.25 0.094 2.18 0.0843 0.0886 0.0988 0.1083 0.1745 0.1209 0.0550
0.6 0.5 0.045 2.03 0.0582 0.0787 0.0710 0.0755 0.1849 0.0763 0.0061
0.75 −0.2 −7.33 212.90 0.4789 0.5543 0.6034 0.6534 0.6786 0.7196 0.6282
0.75 −0.1 −0.69 2.60 0.3719 0.4323 0.4801 0.5309 0.5751 0.6107 0.5143
0.75 0 −0.24 1.95 0.2621 0.3037 0.3553 0.4024 0.4517 0.4827 0.3776
0.75 0.25 0.18 2.12 0.1165 0.1257 0.1473 0.1640 0.2680 0.1927 0.0905
0.75 0.5 0.12 1.99 0.0759 0.1001 0.0943 0.1053 0.2443 0.1145 0.0126
0.9 −0.2 −7.49 229.48 0.5260 0.6127 0.6513 0.6996 0.7457 0.7658 0.6586
0.9 −0.1 −0.49 2.20 0.4261 0.5124 0.5553 0.6099 0.6801 0.6868 0.5609
0.9 0 −0.09 1.84 0.3172 0.3835 0.4262 0.4784 0.5613 0.5623 0.4276
0.9 0.25 0.26 2.11 0.1477 0.1674 0.1954 0.2236 0.3580 0.2680 0.1264
0.9 0.5 0.19 1.97 0.0949 0.1212 0.1220 0.1406 0.3147 0.1610 0.0245
30 0.6 −0.2 −7.26 198.82 0.5934 0.6519 0.7169 0.7668 0.7718 0.8354 0.7499
0.6 −0.1 −0.94 3.29 0.4511 0.4982 0.5795 0.6416 0.6520 0.7308 0.6246
0.6 0 −0.42 2.16 0.3098 0.3333 0.4146 0.4781 0.5053 0.5834 0.4624
0.6 0.25 0.094 2.18 0.1205 0.1212 0.1474 0.1652 0.2653 0.1912 0.0620
0.6 0.5 0.045 2.03 0.0805 0.1122 0.1061 0.1189 0.2861 0.1281 0.0043
0.75 −0.2 −7.33 212.90 0.6659 0.7511 0.7929 0.8412 0.8734 0.8981 0.8023
0.75 −0.1 −0.69 2.60 0.5417 0.6214 0.6830 0.7461 0.7896 0.8241 0.6938
0.75 0 −0.24 1.95 0.4024 0.4666 0.5349 0.6067 0.6774 0.7111 0.5350
0.75 0.25 0.18 2.12 0.1742 0.1840 0.2267 0.2652 0.4118 0.3261 0.1180
0.75 0.5 0.12 1.99 0.1069 0.1420 0.1491 0.1752 0.3886 0.2025 0.0116
0.9 −0.2 −7.49 229.48 0.7231 0.8172 0.8443 0.8878 0.9296 0.9356 0.8355
0.9 −0.1 −0.49 2.20 0.6087 0.7052 0.7479 0.8078 0.8695 0.8784 0.7382
0.9 0 −0.09 1.84 0.4727 0.5774 0.6254 0.6976 0.7937 0.7981 0.6065
0.9 0.25 0.26 2.11 0.2332 0.2681 0.3154 0.3724 0.5553 0.4579 0.1776
0.9 0.5 0.19 1.97 0.1454 0.1909 0.2056 0.2475 0.4999 0.2963 0.0253
50 0.6 −0.2 −7.26 198.82 0.8089 0.8683 0.9064 0.9419 0.9561 0.9758 0.9312
0.6 −0.1 −0.94 3.29 0.6671 0.7373 0.8026 0.8672 0.8979 0.9378 0.8517
0.6 0 −0.42 2.16 0.4852 0.5344 0.6301 0.7168 0.7781 0.8425 0.6888
0.6 0.25 0.094 2.18 0.1834 0.1859 0.2449 0.2908 0.4652 0.3788 0.1042
0.6 0.5 0.045 2.03 0.1144 0.1770 0.1741 0.2184 0.5080 0.2845 0.0188
0.75 −0.2 −7.33 212.90 0.8758 0.9351 0.9514 0.9734 0.9880 0.9916 0.9596
0.75 −0.1 −0.69 2.60 0.7653 0.8572 0.8882 0.9345 0.9669 0.9772 0.9053
0.75 0 −0.24 1.95 0.6089 0.7146 0.7725 0.8515 0.9194 0.9356 0.7887
0.75 0.25 0.18 2.12 0.2825 0.3179 0.3852 0.4729 0.6954 0.6126 0.2144
0.75 0.5 0.12 1.99 0.1714 0.2372 0.2559 0.3257 0.6598 0.4337 0.0422
0.9 −0.2 −7.49 229.48 0.9219 0.9685 0.9734 0.9884 0.9964 0.9974 0.9756
0.9 −0.1 −0.49 2.20 0.8404 0.9238 0.9354 0.9674 0.9895 0.9907 0.9390
0.9 0 −0.09 1.84 0.7120 0.8372 0.8610 0.9229 0.9743 0.9767 0.8601
0.9 0.25 0.26 2.11 0.3854 0.4586 0.5274 0.6340 0.8434 0.7846 0.3480
0.9 0.5 0.19 1.97 0.2419 0.3130 0.3565 0.4512 0.7868 0.5925 0.0852
Table 4

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group Ib.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 0.6 0.6 0 1.99 0.0494 0.0587 0.0519 0.0525 0.1019 0.0508 0.0230
0.6 0.9 −0.14 1.92 0.0608 0.0715 0.0678 0.0704 0.1384 0.0744 0.0321
0.75 0.6 0.07 1.94 0.0514 0.0612 0.0563 0.0583 0.1167 0.0590 0.0258
0.75 0.75 0 1.89 0.0550 0.0690 0.0621 0.0638 0.1283 0.0640 0.0250
0.75 0.9 −0.07 1.86 0.0586 0.0736 0.0661 0.0678 0.1414 0.0693 0.0261
0.9 0.9 0 1.83 0.0594 0.0769 0.0683 0.0709 0.1518 0.0737 0.0269
20 0.6 0.6 0 1.99 0.0614 0.0866 0.0791 0.0871 0.2162 0.0866 0.0053
0.6 0.9 −0.14 1.92 0.0908 0.1216 0.1187 0.1349 0.3255 0.1536 0.0123
0.75 0.6 0.07 1.94 0.0707 0.0994 0.0931 0.1029 0.2637 0.1110 0.0070
0.75 0.75 0 1.89 0.0739 0.1115 0.1018 0.1167 0.3050 0.1303 0.0066
0.75 0.9 −0.07 1.86 0.0852 0.1247 0.1184 0.1387 0.3531 0.1583 0.0088
0.9 0.9 0 1.83 0.0889 0.1339 0.1255 0.1495 0.3830 0.1743 0.0066
30 0.6 0.6 0 1.99 0.0819 0.1255 0.1143 0.1353 0.3322 0.1498 0.0031
0.6 0.9 −0.14 1.92 0.1267 0.1803 0.1876 0.2284 0.5056 0.2760 0.0124
0.75 0.6 0.07 1.94 0.1010 0.1483 0.1440 0.1755 0.4253 0.2066 0.0063
0.75 0.75 0 1.89 0.1079 0.1763 0.1654 0.2035 0.4877 0.2465 0.0058
0.75 0.9 −0.07 1.86 0.1266 0.1955 0.1931 0.2407 0.5529 0.2966 0.0094
0.9 0.9 0 1.83 0.1304 0.2121 0.2101 0.2660 0.6005 0.3292 0.0105
50 0.6 0.6 0 1.99 0.1228 0.2037 0.1969 0.2546 0.5836 0.3422 0.0204
0.6 0.9 −0.14 1.92 0.2169 0.3165 0.3411 0.4448 0.8110 0.5892 0.0777
0.75 0.6 0.07 1.94 0.1653 0.2574 0.2627 0.3424 0.7086 0.4626 0.0455
0.75 0.75 0 1.89 0.1770 0.2964 0.3009 0.3944 0.7868 0.5399 0.0577
0.75 0.9 −0.07 1.86 0.2163 0.3457 0.3596 0.4695 0.8470 0.6249 0.0908
0.9 0.9 0 1.83 0.2273 0.3696 0.3847 0.5076 0.8852 0.6821 0.1092
Table 5

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group Ic.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 1.25 −0.2 −8.16 280.37 0.3265 0.3885 0.4109 0.4369 0.4507 0.4708 0.4064
1.25 −0.1 −0.14 1.81 0.2700 0.3236 0.3481 0.3722 0.3875 0.4037 0.3448
1.25 0 0.20 1.80 0.2153 0.2567 0.2775 0.2998 0.3207 0.3293 0.2718
1.25 0.25 0.44 2.17 0.1251 0.1417 0.1524 0.1630 0.2148 0.1792 0.1259
1.25 0.5 0.35 2.00 0.0841 0.1006 0.1002 0.1080 0.1861 0.1159 0.0609
1.25 0.75 0.218 1.86 0.0715 0.0907 0.0852 0.0892 0.1768 0.0944 0.0394
1.25 0.95 0.12 1.80 0.0658 0.0849 0.0793 0.0846 0.1808 0.0934 0.0348
2 −0.2 −10.64 443.42 0.3924 0.4727 0.4942 0.5218 0.5579 0.5563 0.4749
2 −0.1 0.376 1.85 0.3283 0.3984 0.4228 0.4486 0.4854 0.4824 0.4028
2 0 0.64 2.14 0.2761 0.3349 0.3562 0.3815 0.4211 0.4168 0.3395
2 0.25 0.75 2.5 0.1712 0.2042 0.2191 0.2372 0.2952 0.2598 0.1867
2 0.5 0.61 2.24 0.1190 0.1389 0.1423 0.1535 0.2292 0.1651 0.0948
2 0.75 0.46 2.02 0.0879 0.1096 0.1082 0.1153 0.2067 0.1249 0.0559
2 0.95 0.36 1.91 0.0773 0.0965 0.0923 0.1000 0.1968 0.1075 0.0445
20 1.25 −0.2 −8.16 280.37 0.6143 0.7226 0.7500 0.7978 0.8623 0.8521 0.7284
1.25 −0.1 −0.14 1.81 0.5149 0.6280 0.6565 0.7136 0.8017 0.7829 0.6353
1.25 0 0.20 1.80 0.4177 0.5265 0.5611 0.6216 0.7316 0.7043 0.5273
1.25 0.25 0.44 2.17 0.2315 0.2795 0.3133 0.3560 0.5276 0.4262 0.2121
1.25 0.5 0.35 2.00 0.1496 0.1876 0.1988 0.2314 0.4470 0.2694 0.0556
1.25 0.75 0.218 1.86 0.1125 0.1614 0.1627 0.1934 0.4370 0.2281 0.188
1.25 0.95 0.12 1.80 0.1033 0.1587 0.1505 0.1805 0.4494 0.2152 0.109
2 −0.2 −10.64 443.42 0.7103 0.8211 0.8345 0.8724 0.9264 0.9137 0.7930
2 −0.1 0.376 1.85 0.6275 0.7484 0.7664 0.8142 0.8888 0.8698 0.7169
2 0 0.64 2.14 0.5395 0.6675 0.6915 0.7463 0.8423 0.8153 0.6311
2 0.25 0.75 2.5 0.3402 0.4260 0.4578 0.5154 0.6812 0.6018 0.3391
2 0.5 0.61 2.24 0.2205 0.2696 0.2957 0.3343 0.5493 0.3945 0.1197
2 0.75 0.46 2.02 0.1650 0.2104 0.2240 0.2613 0.5116 0.3073 0.0461
2 0.95 0.36 1.91 0.1330 0.1869 0.1913 0.2254 0.4948 0.2663 0.0251
30 1.25 −0.2 −8.16 280.37 0.8203 0.9067 0.9152 0.9446 0.9756 0.9716 0.8903
1.25 −0.1 −0.14 1.81 0.7230 0.8401 0.8535 0.8989 0.9520 0.9467 0.8201
1.25 0 0.20 1.80 0.6124 0.7423 0.7692 0.8332 0.9193 0.9054 0.7202
1.25 0.25 0.44 2.17 0.3549 0.4479 0.4875 0.5646 0.7556 0.6759 0.3159
1.25 0.5 0.35 2.00 0.2314 0.2950 0.3263 0.3921 0.6730 0.4717 0.0755
1.25 0.75 0.218 1.86 0.1726 0.2546 0.2684 0.3310 0.6716 0.4151 0.0257
1.25 0.95 0.12 1.80 0.1699 0.2634 0.2668 0.3406 0.6981 0.4303 0.0195
2 −0.2 −10.64 443.42 0.8910 0.9546 0.9577 0.9751 0.9915 0.9886 0.9317
2 −0.1 0.376 1.85 0.8347 0.9240 0.9288 0.9545 0.9852 0.9785 0.8916
2 0 0.64 2.14 0.7498 0.8671 0.8747 0.9204 0.9680 0.9616 0.8240
2 0.25 0.75 2.5 0.5037 0.6379 0.6692 0.7457 0.8844 0.8365 0.5026
2 0.5 0.61 2.24 0.3409 0.4246 0.4733 0.5489 0.7872 0.6398 0.1726
2 0.75 0.46 2.02 0.2594 0.3369 0.3739 0.4488 0.7409 0.5351 0.0647
2 0.95 0.36 1.91 0.2114 0.3039 0.3276 0.4058 0.7343 0.4959 0.0402
50 1.25 −0.2 −8.16 280.37 0.9685 0.9912 0.9912 0.9963 0.9997 0.9995 0.9882
1.25 −0.1 −0.14 1.81 0.9328 0.9800 0.9814 0.9922 0.9989 0.9985 0.9770
1.25 0 0.20 1.80 0.8616 0.9483 0.9512 0.9787 0.9967 0.9958 0.9385
1.25 0.25 0.44 2.17 0.5714 0.7194 0.7429 0.8415 0.9628 0.9390 0.5867
1.25 0.5 0.35 2.00 0.3989 0.5047 0.5584 0.6732 0.9212 0.8031 0.2253
1.25 0.75 0.218 1.86 0.3115 0.4453 0.4843 0.6161 0.9284 0.7756 0.1641
1.25 0.95 0.12 1.80 0.2931 0.4568 0.4857 0.6223 0.9430 0.7948 0.1778
2 −0.2 −10.64 443.42 0.9908 0.9984 0.9977 0.9996 0.9999 0.9999 0.9962
2 −0.1 0.376 1.85 0.9783 0.9955 0.9959 0.9985 0.9999 0.9999 0.9924
2 0 0.64 2.14 0.9497 0.9869 0.9864 0.9963 0.9996 0.9995 0.9775
2 0.25 0.75 2.5 0.7702 0.8976 0.9009 0.9541 0.9924 0.9883 0.8129
2 0.5 0.61 2.24 0.5746 0.7035 0.7426 0.8353 0.9720 0.9262 0.4443
2 0.75 0.46 2.02 0.4377 0.5643 0.6216 0.7389 0.9589 0.8649 0.2695
2 0.95 0.36 1.91 0.3741 0.5193 0.5706 0.7036 0.9585 0.8503 0.2320
Table 6

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group II.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 1.25 1.25 0 1.76 0.0663 0.0877 0.0785 0.0857 0.1919 0.0911 0.0317
1.25 1.5 −0.09 1.76 0.0713 0.0904 0.0828 0.0894 0.1918 0.0954 0.0341
1.25 1.75 −0.17 1.79 0.0732 0.0948 0.0888 0.0939 0.1982 0.0999 0.0366
1.5 1.5 0 1.75 0.0697 0.0933 0.0851 0.0911 0.2000 0.0966 0.0313
1.75 1.5 0.08 1.76 0.0703 0.0906 0.0834 0.0914 0.1943 0.0971 0.0315
1.75 1.75 0 1.77 0.0677 0.0888 0.0805 0.0870 0.1889 0.0923 0.0312
20 1.25 1.25 0 1.76 0.1127 0.1743 0.1656 0.2022 0.4926 0.244 0.0116
1.25 1.5 −0.09 1.76 0.1104 0.1723 0.1665 0.2051 0.4929 0.2481 0.0107
1.25 1.75 −0.17 1.79 0.1168 0.1755 0.1735 0.2127 0.4990 0.2559 0.0125
1.5 1.5 0 1.75 0.1119 0.1811 0.1727 0.2101 0.5066 0.2536 0.0093
1.75 1.5 0.08 1.76 0.1127 0.1736 0.1663 0.2027 0.4934 0.2470 0.0109
1.75 1.75 0 1.77 0.1059 0.1660 0.1593 0.1960 0.4770 0.2345 0.0093
30 1.25 1.25 0 1.76 0.1737 0.2781 0.2825 0.3595 0.7270 0.4581 0.0220
1.25 1.5 −0.09 1.76 0.1798 0.2898 0.2985 0.3775 0.7436 0.4783 0.0235
1.25 1.75 −0.17 1.79 0.1812 0.2841 0.2954 0.3744 0.7354 0.4714 0.0246
1.5 1.5 0 1.75 0.1773 0.2880 0.2964 0.3777 0.7462 0.4778 0.0238
1.75 1.5 0.08 1.76 0.1705 0.2716 0.2763 0.3538 0.7280 0.4543 0.0200
1.75 1.75 0 1.77 0.1705 0.2705 0.2752 0.3495 0.7169 0.4409 0.0186
50 1.25 1.25 0 1.76 0.3023 0.4837 0.5126 0.6588 0.9572 0.8309 0.2068
1.25 1.5 −0.09 1.76 0.3199 0.4928 0.5231 0.6727 0.9641 0.8387 0.2224
1.25 1.75 −0.17 1.79 0.3294 0.5013 0.5359 0.6825 0.9621 0.8443 0.2245
1.5 1.5 0 1.75 0.3131 0.4922 0.5196 0.6689 0.9640 0.8392 0.2176
1.75 1.5 0.08 1.76 0.3053 0.4854 0.5174 0.6570 0.9581 0.8294 0.2104
1.75 1.75 0 1.77 0.2948 0.4714 0.5000 0.6467 0.9504 0.8143 0.1966
Table 7

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group III.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 2.5 1.25 0.35 1.92 0.0685 0.0869 0.0815 0.0864 0.1820 0.0932 0.0369
2.5 1.5 0.25 1.87 0.0672 0.0857 0.0752 0.0795 0.1754 0.0853 0.0311
2.5 1.75 0.18 1.86 0.0575 0.0747 0.0687 0.0714 0.1590 0.0736 0.0261
3 1.25 0.44 2.03 0.0661 0.0825 0.0783 0.0809 0.1677 0.0854 0.0323
3 1.25 0.34 1.97 0.0604 0.0741 0.0690 0.0727 0.1540 0.0774 0.0277
3 1.75 0.26 1.95 0.0586 0.0724 0.0648 0.0678 0.1481 0.0694 0.0246
5 1.25 0.67 2.50 0.0530 0.0661 0.0601 0.0621 0.1343 0.0635 0.0263
5 1.5 0.56 2.39 0.0532 0.0645 0.0576 0.0604 0.1273 0.0615 0.0236
5 1.75 0.46 2.34 0.0474 0.0565 0.0507 0.0515 0.1151 0.0532 0.0219
20 2.5 1.25 0.35 1.92 0.1145 0.1654 0.1603 0.1939 0.4554 0.2313 0.0131
2.5 1.5 0.25 1.87 0.0952 0.1449 0.1416 0.1711 0.4300 0.2030 0.0099
2.5 1.75 0.18 1.86 0.0942 0.1404 0.1342 0.1583 0.3993 0.1871 0.0075
3 1.25 0.44 2.03 0.1022 0.1504 0.1439 0.1733 0.4118 0.2042 0.0121
3 1.25 0.34 1.97 0.0902 0.1341 0.1265 0.1489 0.3877 0.1742 0.0083
3 1.75 0.26 1.95 0.0775 0.1186 0.1091 0.1303 0.3477 0.1504 0.0064
5 1.25 0.67 2.50 0.0758 0.1131 0.1030 0.1213 0.3225 0.1388 0.0064
5 1.5 0.56 2.39 0.0680 0.1015 0.0907 0.1052 0.2980 0.1203 0.0052
5 1.75 0.46 2.34 0.0681 0.0938 0.0865 0.1010 0.2751 0.1143 0.0065
30 2.5 1.25 0.35 1.92 0.1738 0.2648 0.2717 0.3435 0.6897 0.4327 0.0213
2.5 1.5 0.25 1.87 0.1564 0.2444 0.2474 0.3072 0.6673 0.3904 0.0160
2.5 1.75 0.18 1.86 0.1418 0.2238 0.2234 0.2811 0.6302 0.3560 0.0132
3 1.25 0.44 2.03 0.1556 0.2317 0.2403 0.3040 0.6417 0.3813 0.0183
3 1.25 0.34 1.97 0.1342 0.2117 0.2111 0.2672 0.6145 0.3388 0.0116
3 1.75 0.26 1.95 0.1154 0.1876 0.1839 0.2333 0.5668 0.2941 0.0080
5 1.25 0.67 2.50 0.1119 0.1726 0.1676 0.2126 0.5252 0.2703 0.0067
5 1.5 0.56 2.39 0.0959 0.1552 0.1465 0.1855 0.4937 0.2362 0.0056
5 1.75 0.46 2.34 0.0966 0.1460 0.1442 0.1758 0.4481 0.2154 0.0066
50 2.5 1.25 0.35 1.92 0.3061 0.4544 0.4933 0.6279 0.9408 0.7960 0.1815
2.5 1.5 0.25 1.87 0.2696 0.4270 0.4552 0.5921 0.9287 0.7646 0.1541
2.5 1.75 0.18 1.86 0.2345 0.3853 0.4065 0.5364 0.9041 0.7130 0.1246
3 1.25 0.44 2.03 0.2677 0.4100 0.4421 0.5778 0.9203 0.7560 0.1402
3 1.25 0.34 1.97 0.2221 0.3669 0.3821 0.5091 0.8971 0.6924 0.1115
3 1.75 0.26 1.95 0.1933 0.3213 0.3346 0.4537 0.8640 0.6332 0.0836
5 1.25 0.67 2.50 0.1781 0.2920 0.3015 0.4102 0.8381 0.5889 0.0675
5 1.5 0.56 2.39 0.1568 0.2665 0.2699 0.3702 0.8056 0.5374 0.0509
5 1.75 0.46 2.34 0.1555 0.2471 0.2532 0.3415 0.7590 0.4906 0.0438
Table 8

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group IV.

n λ3 λ4 α3 α4 D V CH A2 KLmn W JB
10 2.5 2.5 0 1.91 0.0508 0.0634 0.0574 0.0593 0.1235 0.0605 0.0247
2.5 3 −0.08 1.98 0.0462 0.0547 0.0487 0.0497 0.1086 0.0494 0.0215
2.5 4 −0.2 2.17 0.0462 0.0508 0.0469 0.0478 0.0874 0.0480 0.0240
2.5 10 −0.33 3.02 0.1550 0.1582 0.1775 0.1808 0.1764 0.1827 0.1225
3 2.5 0.08 1.98 0.0475 0.0529 0.0482 0.0495 0.1033 0.0506 0.0224
3 3 0 2.06 0.0394 0.0459 0.0413 0.0414 0.0874 0.0419 0.0200
3 4 −0.11 2.24 0.0437 0.0468 0.0434 0.0422 0.0673 0.0417 0.0262
3 5 −0.18 2.44 0.0556 0.0543 0.0565 0.0548 0.0675 0.0534 0.0350
3 10 −0.22 3.14 0.2118 0.2105 0.2384 0.2394 0.1953 0.2377 0.1732
5 2.5 0.27 2.36 0.0591 0.0603 0.0588 0.0584 0.0859 0.0571 0.0331
5 3 0.18 2.44 0.0592 0.0576 0.0580 0.0564 0.0699 0.0544 0.0361
5 4 0.06 2.66 0.0650 0.0679 0.0661 0.0619 0.0501 0.0553 0.0477
5 5 0 2.90 0.0837 0.0877 0.0889 0.0834 0.0459 0.0711 0.0657
5 10 −0.007 3.83 0.3532 0.3628 0.3867 0.3711 0.2141 0.3341 0.2725
20 2.5 2.5 0 1.91 0.0671 0.1027 0.0921 0.1078 0.2980 0.1243 0.0054
2.5 3 −0.08 1.98 0.0584 0.0814 0.0722 0.0825 0.2372 0.0915 0.0048
2.5 4 −0.2 2.17 0.0537 0.0686 0.0602 0.0655 0.1755 0.0692 0.0076
2.5 10 −0.33 3.02 0.3245 0.3243 0.4015 0.4188 0.4262 0.4170 0.1666
3 2.5 0.08 1.98 0.0586 0.0811 0.0720 0.0818 0.2320 0.0898 0.0042
3 3 0 2.06 0.0430 0.0595 0.0515 0.0563 0.1750 0.0619 0.0046
3 4 −0.11 2.24 0.0421 0.0518 0.0440 0.0467 0.1169 0.0457 0.0068
3 5 −0.18 2.44 0.0715 0.0766 0.0832 0.0825 0.1160 0.0737 0.0226
3 10 −0.22 3.14 0.4228 0.4102 0.4996 0.5063 0.4310 0.4862 0.2540
5 2.5 0.27 2.36 0.0713 0.0814 0.0884 0.0930 0.1732 0.0924 0.0167
5 3 0.18 2.44 0.0723 0.0755 0.0800 0.0823 0.1122 0.0729 0.0200
5 4 0.06 2.66 0.0802 0.0853 0.0851 0.0778 0.0613 0.0570 0.0263
5 5 0 2.90 0.1200 0.1433 0.1333 0.1117 0.1350 0.0655 0.0370
5 10 −0.007 3.83 0.6405 0.6593 0.7047 0.6772 0.4580 0.5857 0.4217
30 2.5 2.5 0 1.91 0.0963 0.1551 0.1461 0.1823 0.4836 0.2291 0.0060
2.5 3 −0.08 1.98 0.0792 0.1202 0.1101 0.1334 0.3929 0.1659 0.0027
2.5 4 −0.2 2.17 0.0672 0.0924 0.0868 0.1024 0.2960 0.1197 0.0033
2.5 10 −0.33 3.02 0.4937 0.4925 0.6021 0.6278 0.6249 0.6310 0.2283
3 2.5 0.08 1.98 0.0723 0.1140 0.1088 0.1361 0.3959 0.1684 0.0039
3 3 0 2.06 0.0568 0.0857 0.0759 0.0924 0.3040 0.1114 0.0023
3 4 −0.11 2.24 0.0513 0.0665 0.0623 0.0680 0.2101 0.0718 0.0044
3 5 −0.18 2.44 0.1012 0.1044 0.1156 0.1205 0.2059 0.1094 0.0158
3 10 −0.22 3.14 0.6192 0.5951 0.7129 0.7219 0.6376 0.7009 0.3593
5 2.5 0.27 2.36 0.1054 0.1189 0.1293 0.1431 0.2887 0.1505 0.0123
5 3 0.18 2.44 0.0979 0.1009 0.1140 0.1181 0.2093 0.1081 0.0159
5 4 0.06 2.66 0.1013 0.1124 0.1111 0.1022 0.1351 0.0669 0.0154
5 5 0 2.90 0.1508 0.1838 0.1678 0.1504 0.1413 0.0796 0.0224
5 10 −0.007 3.83 0.8310 0.8439 0.8816 0.8601 0.7556 0.7767 0.5465
50 2.5 2.5 0 1.91 0.1472 0.2572 0.2582 0.3570 0.7916 0.5213 0.0492
2.5 3 −0.08 1.98 0.1124 0.1926 0.1830 0.2580 0.7001 0.3961 0.0254
2.5 4 −0.2 2.17 0.0952 0.1453 0.1387 0.1894 0.5684 0.2821 0.0128
2.5 10 −0.33 3.02 0.7504 0.7489 0.8554 0.8832 0.8954 0.9051 0.5397
3 2.5 0.08 1.98 0.1121 0.1932 0.1847 0.2604 0.6965 0.3970 0.0245
3 3 0 2.06 0.0723 0.1236 0.1121 0.1580 0.5702 0.2573 0.0090
3 4 −0.11 2.24 0.0617 0.0873 0.0805 0.1074 0.4141 0.1532 0.0045
3 5 −0.18 2.44 0.1514 0.1622 0.1831 0.2023 0.4169 0.2204 0.0179
3 10 −0.22 3.14 0.8553 0.8323 0.9188 0.9295 0.9015 0.9295 0.6700
5 2.5 0.27 2.36 0.1540 0.1875 0.2103 0.2590 0.5529 0.3328 0.0259
5 3 0.18 2.44 0.1477 0.1598 0.1794 0.2019 0.4190 0.2214 0.0186
5 4 0.06 2.66 0.1456 0.1603 0.1600 0.1536 0.2932 0.1068 0.0089
5 5 0 2.90 0.2347 0.3042 0.2761 0.2515 0.3472 0.1249 0.0086
5 10 −0.007 3.83 0.9684 0.9720 0.9845 0.9796 0.9629 0.9555 0.7671
Table 9

Power comparisons of the tests at significance level 0.05 for sample sizes n = 10, 20, 30 and 50 under alternatives from group V.

From Tables 39, we can see that the power values of the tests are considerably different. We can select the tests which are, generally, the most powerful against the alternatives from the given groups.

In group Ia, it is seen that the tests JB and W generally have the most power and the test KLmn has the most power against the symmetric alternatives with parameters λ3=λ4=0.25,0.5. The difference powers between JB (or W) with the other tests are considerable.

In group Ib, sometimes the test W gives a higher power and sometimes the test KLmn does. When the sample size increases the difference power values become substantial. It is seen that, in groups Ia and Ib, when λ4>0, the test KLmn is powerful and otherwise the test W is powerful.

In group Ic, a uniform superiority of KLmn test to other tests is obvious. The difference of power values between the KLmn test and the other tests is substantial. We note that in this group(Ic) 0.5<λ4<1.

In group II, the KLmn test has the most power and the other tests have low powers. The difference power values between the KLmn test and the other tests are substantial. When λ4<0, we can see that the difference power values between KLmn and W are small.

In groups III and IV, since λ4>0 the KLmn test again has the most power. The difference powers between the KLmn test and the other tests are substantial. For these groups, the test KLmn, based on the sample entropy, has very good power values but the other tests have low powers.

Finally, in group V, the test based on KLmn statistic has generally the most power. We can see that when λ4>5, the KLmn test dose not achieve the most power, but when sample size increases the power of KLmn test increases and the difference powers between KLmn test and the other tests become small. Therefore, we can conclude that the KLmn test is powerful in this group.

Briefly, the best test in term of power for different groups is presented in Table 10.

Groups (Alternatives)
I II III IV V
Ia Ib Ic
JB, W W, KLmn KLmn KLmn KLmn KLmn KLmn
Table 10

The best test in term of power performance in different groups.

It should be noted that the KS and JB tests have the least power values in groups Ia, Ib, II and Ic, III, IV, V, respectively.

4. ILLUSTRATION WITH SOME REAL DATA

In this section, we apply some real data examples to show the behavior of the tests. The histograms of the considered data sets are displayed in Figs. 8 and 9.

Figure 8

Histograms for data in Examples 1 and 2.

Figure 9

Histogram for data set in Example 3.

Example 1.

The following data are 100 breaking strengths of yarn presented by Duncan [21]:

66, 117, 132, 111, 107, 85, 89, 79, 91, 97, 138, 103, 111, 86, 78, 96, 93, 101, 102, 110, 95, 96, 88, 122, 115, 92, 137, 91, 84, 96, 97, 100, 105, 104, 137, 80, 104, 104, 106, 84, 92, 86, 104, 132, 94, 99, 102, 101, 104, 107, 99, 85, 95, 89, 102, 100, 98, 97, 104, 114, 111, 98, 99, 102, 91, 95, 111, 104, 97, 98, 102, 109, 88, 91, 103, 94, 105, 103, 96, 100, 101, 98, 97, 97, 101, 102, 98, 94, 100, 98, 99, 92, 102, 87, 99, 62, 92, 100, 96, 98.

Puig and Stephens [22] used the Empirical distribution function (EDF) tests for fitting a normal distribution for these data. They concluded that all the test statistics have significance levels below 0.01 so that the normal assumption is rejected at this level of significance.

We apply the tests for testing the assumption that the data are from a normal distribution. We obtain the maximum likelihood estimator of FMKL family as

λ^1=98.6370,λ^2=0.2715,λ^3=0.2678,λ^4=0.3866.

Since λ^3,λ^40.5, density of these data belong to class Ia. Based on our simulations, in this class generally the test JB has the most power and we should use this test. Moreover, if we assume approximately λ^3λ^4, the test based on KLmn is appreciate.

The values of the test statistics are

D=0.1569,V=0.2462,CH=0.4503,A2=2.7180,KLmn=0.1887,W=0.9904,JB=31.5526,
and then the p-values obtained are tabulated in Table 11.

Test Value of the test statistic p-value
D 0.1569 0.0000
V 0.2462 0.0000
CH 0.4503 0.0000
A2 2.7180 0.0000
KLmn 0.1887 0.0006
W 0.9904 0.0000
JB 31.5526 0.0009
Table 11

Comparison of p-values of the tests in Example 1.

Therefore, the normal assumption is rejected by the EDF statistics. Also, the tests KLmn, W and JB reject the normal assumption.

Example 2.

Bain and Engelhardt [23] presented the following dataset, consisting of 33 difference in flood levels between stations on a river:

1.96, 1.97, 3.60, 3.80, 4.79, 5.66, 5.76, 5.78, 6.27, 6.30,6.76, 7.65, 7.84, 7.99, 8.51, 9.18, 10.13, 10.24, 10.25, 10.43, 11.45, 11.48, 11.75, 11.81, 12.34, 12.78, 13.06, 13.29, 13.98, 14.18, 14.40, 16.22, 17.06.

They suggested that the Laplace distribution might provide a good fit. Puig and Stephens [22] concluded that D and W2 reject the Laplace distribution for the data at 0.05 level and the other tests accept the Laplace assumption.

We apply the tests for testing the assumption that the data are from a normal distribution. We obtain the maximum likelihood estimator of FMKL family as

λ^1=8.0967,λ^2=0.1564,λ^3=0.9997,λ^4=0.6779.

Since 0.5<λ^3,λ^4<1, density of these data belong to class Ic. Based on our simulations, in this class the test KLmn has the most power and we should use this test.

The values of the test statistics are

D=0.0929,V=0.1722,CH=0.0416,A2=0.2467,KLmn=0.2175,W=0.9766,JB=1.3645,
and then the p-values are obtained and presented in Table 12.

Test Value of the test statistic p-value
D 0.0929 0.6599
V 0.1722 0.5707
CH 0.0416 0.6455
A2 0.2467 0.7489
KLmn 0.2175 0.3734
W 0.9766 0.6798
JB 1.3645 0.3857
Table 12

Comparison of p-values of the tests in Example 2.

Therefore, the normal assumption is accepted by all tests at the significance level of 0.05. Based on our simulation results presented in Section 3, we can accept the result obtained by KLmn test.

Example 3.

The following data represent active repair times (in hours) for an airborne communication transceiver:

0.2, 0.3, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.7, 0.8, 0.8, 1.0, 1.0, 1.0, 1.0, 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.5, 3.0, 3.0, 3.3, 3.3, 4.0, 4.0, 4.5, 4.7, 5.0, 5.4, 5.4, 7.0, 7.5, 8.8, 9.0, 10.3, 22.0, 24.5.

von Alven [24] fitted the lognormal distribution for these data. Chhikara and Folks [25] fitted the Inverse Gaussian (IG) distribution and justified it by the Kolmogorov–Smirnov statistic. Finally, Lee et al. [26] tested the lognormal and IG distributions which were both accepted.

We obtain the maximum likelihood estimator of FMKL family as

λ^1=1.2184,λ^2=0.6639,λ^3=1.4792,λ^4=0.6779.

Since λ^3>1 and λ^4<1, density of these data belong to class II. Based on our simulations, in this class the test based on KLmn statistic has the most power and we should use this test.

In this case, we obtained the values of the test statistics for normal model as

D=0.2465,V=0.4614,CH=0.8793,A2=5.0138,KLmn=0.9404,W=0.6317,JB=177.19,
and then the p-values are obtained and presented in Table 13.

Test Value of the test statistic p-value
D 0.2465 0.0000
V 0.4614 0.0000
CH 0.8793 0.0000
A2 5.0138 0.0000
KLmn 0.9404 0.0000
W 0.6317 0.0000
JB 177.19 0.0000
Table 13

Comparison of p-values of the tests in Example 2.

Therefore, the normal assumption is rejected by all tests at the significance level of 0.05. Based on our power study, we accept the result obtained by KLmn test.

5. CONCLUSIONS

In this paper, we considered seven popular tests for normality, namely, Kolmogorov-Smirnov, Anderson-Darling, Kuiper, Jarque-Bera, Cramer von Mises, Shapiro-Wilk and Vasicek. These tests are commonly used in practice and statistical software and therefore power values of these tests against various alternatives are important. Here, we considered the family of four-parameter GLDs which is called FMKL family, as alternatives for tests for normality.

The FMKL family is divided to five groups and therefore we computed the power values of the tests against these five groups using Monte Carlo computations for sample sizes n = 10, 20, 30 and n = 50. Differences in power of the seven tests are considerable and each of the tests JB, W and KLmn can be most powerful depending on the type of alternatives. In group I, unimodal densities with continuous tails, the tests JB and W have the most power and when λ3=λ4 (symmetric distribution) or λ4>0 the test KLmn has the most power. In groups II-V, monotone, U-shaped, S-shaped densities and unimodal pdfs with both tail truncated, the KLmn test generally is most powerful.

Based on these observations, we can formulate the following recommendations for the application of the tests in practice.

  1. Use the statistics JB or W, if the assumed alternatives are in groups Ia and Ib with the exception of the cases where λ3=λ4 (symmetric distribution) and λ4>0.

  2. Use the statistic KLmn, based on the sample entropy, if the assumed alternatives are not in groups Ia and Ib with the exception of the cases where λ3=λ4 (symmetric distribution) and λ4>0.

The author is thankful to the referees and editor-in-chief of the journal for their valuable comments and suggestions, which improved the presentation of this paper greatly.

Footnotes

Declarations: The author has any competing interests in the manuscript.

REFERENCES

1.J.S. Ramberg and B.W. Schmeiser, Commun. ACM., Vol. 17, 1974, pp. 78-82.
2.M. Freimer, G. Mudholkar, G. Kollia, and C. Lin, Commun. Stat. Theory Methods., Vol. 17, 1988, pp. 3547-3567.
3.R.B. D'Agostino and M.A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker, Inc,, New York, 1986.
4.C. Huber-Carol, N. Balakrishnan, M.S. Nikulin, and M. Mesbah, Goodness-of-Fit Tests and Model Validity, Birkhäuser, Boston, Basel, Berlin, 2002.
5.H. Thode Jr., Testing for Normality, Marcel Dekker, New York, 2002.
6.Alizadeh H. Noughabi and N.R. Arghami, J.Stat. Comput. Simul., Vol. 81, 2011, pp. 965-972.
7.A. Harri and K.H. Coble, J. Appl.Stat., Vol. 38, 2011, pp. 1369-1379.
8.J.A.T. Sanqui, T.T. Nguyen, and A.K. Gupta, J.Stat. Comput. Simul., Vol. 82, 2012, pp. 359-368.
9.E. Zamanzade and N.R. Arghami, J.Stat. Comput. Simul., Vol. 82, 2012, pp. 1701-1713.
10.F. Marmolejo-Ramos and J. González-Burgos, Methodology, Vol. 9, 2013, pp. 137-149.
11.D.W. Joenssen and J. Vogel, J.Stat. Comput. Simul., Vol. 84, 2014, pp. 1055-1078.
12.C.C. Wang, J.Stat.Comput. Simul., Vol. 85, 2015, pp. 166-188.
13.R. von Mises, Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik, Deuticke, Leipzig and Vienna, 1931.
14.A.N. Kolmogorov, Giornale dell'Intituto Italiano degli Attuari, Vol. 4, 1933, pp. 83-91.
15.T.W. Anderson and D.A. Darling, J.Am. Stat. Assoc., Vol. 49, 1954, pp. 765-769.
16.N.H. Kuiper, Test concerning random points on a circle, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, series A, Vol. 63, 1962, pp. 38-47.
17.S.S. Shapiro and M.B. Wilk, Biometrika, Vol. 52, 1965, pp. 591-611.
18.O. Vasicek, J.R. Stat. Soc. Ser B., Vol. 38, 1976, pp. 54-59.
19.C.M. Jarque and A.K. Bera, Int. Stat. Rev., Vol. 55, 1987, pp. 163-172.
20.E.S. Pearson and H.O. Hartley, Biometrika Tables for Statisticians, Cambridge University Press, 1972.
21.A.J. Duncan, Quality Control and Industrial Statistics, Irwin, Homewood, 1974.
22.P. Puig and M.A. Stephens, Technometrics, Vol. 42, 2000, pp. 417-424.
23.L.J. Bain and M. Englehardt, Technometrics, Vol. 15, 1973, pp. 875-887.
24.W.H. von Alven, Reliability Engineering by ARINC, Prentice-Hall, New Jersey, 1964.
25.R.S. Chhikara and J.L. Folks, Technometrics, Vol. 19, 1977, pp. 461-468.
26.S. Lee, I. Vonta, and A. Karagrigoriou, Comput. Stat. Data Anal., Vol. 55, 2011, pp. 2635-2643.
Journal
Journal of Statistical Theory and Applications
Volume-Issue
17 - 4
Pages
647 - 660
Publication Date
2018/12/31
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.2018.17.4.7How to use a DOI?
Copyright
© 2018 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Hadi Alizadeh Noughabi
PY  - 2018
DA  - 2018/12/31
TI  - A Comprehensive Study on Power of Tests for Normality
JO  - Journal of Statistical Theory and Applications
SP  - 647
EP  - 660
VL  - 17
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.4.7
DO  - 10.2991/jsta.2018.17.4.7
ID  - Noughabi2018
ER  -