A Simple Empirical Likelihood Ratio Test for Normality Based on the Moment Constraints of a Half-Normal Distribution
Table 6
Results of the Monte Carlo power comparisons based on samples with sizes from symmetric alternative distributions defined on at .
Symmetric alternative distributions defined on at
Distribution
AD
CVM
JB
SW
DB
SEELR
Beta(2,2)
20
0.0564
0.0544
0.0594
0.0052
0.0516
0.1310
0.0696
0.0970
30
0.0786
0.0520
0.0812
0.0012
0.0768
0.2004
0.0550
0.1962
50
0.1222
0.0852
0.1172
0.0010
0.1528
0.3468
0.0628
0.4252
80
0.2340
0.1256
0.1834
0.0128
0.3170
0.5978
0.1128
0.7204
Beta(3,3)
20
0.0404
0.0474
0.0408
0.0076
0.0372
0.0780
0.0518
0.0620
30
0.0786
0.0520
0.0812
0.0046
0.0768
0.1112
0.0392
0.1030
50
0.0736
0.0524
0.0650
0.0014
0.0682
0.1654
0.0326
0.1906
80
0.1076
0.0762
0.0826
0.0022
0.1128
0.2772
0.0298
0.3458
Beta(0.5,0.5)
20
0.6160
0.3098
0.5058
0.0066
0.7190
0.9094
0.7092
0.7015
30
0.8576
0.4998
0.7332
0.0052
0.9392
0.9914
0.8830
0.8960
50
0.9902
0.7976
0.9568
0.3822
0.9992
1.0000
0.9916
0.9956
80
1.0000
0.9724
0.9990
0.9872
1.0000
1.0000
1.0000
1.0000
Uniform(0,1)
20
0.1640
0.1014
0.1396
0.0040
0.1886
0.4064
0.2598
0.3332
30
0.3004
0.1422
0.2262
0.0020
0.3894
0.6622
0.3202
0.6002
50
0.5780
0.2532
0.4282
0.0118
0.7546
0.9358
0.5624
0.9120
80
0.8636
0.4578
0.7092
0.3706
0.9688
0.9990
0.8730
0.9944
Logit-norm(0,1)
20
0.0648
0.0442
0.0562
0.0056
0.0578
0.1294
0.0700
0.1010
30
0.0858
0.0574
0.0748
0.0024
0.0796
0.1974
0.0658
0.1990
50
0.1394
0.0812
0.1220
0.0010
0.1612
0.3420
0.0676
0.4156
80
0.2630
0.1368
0.2114
0.0126
0.3408
0.5830
0.1094
0.7108
Logit-norm(0,2)
20
0.3758
0.1844
0.2934
0.0046
0.4366
0.7034
0.4806
0.5348
30
0.6092
0.2884
0.4822
0.0030
0.7342
0.9150
0.6512
0.8258
50
0.9016
0.5412
0.7814
0.1174
0.9742
0.9976
0.9006
0.9818
80
0.9942
0.8170
0.9644
0.8594
1.0000
1.0000
0.9958
0.9996
Anderson-Darling () test, Modified Kolmogorov-Smirnov () test [2], Cramer-von Mises test () test, Jarque-Bera () test, Shapiro-Wilk () test, density based empirical likelihood ratio based () test [16], simple and exact empirical likelihood ratio based () test [13], and the proposed test .
