A Simple Empirical Likelihood Ratio Test for Normality Based on the Moment Constraints of a Half-Normal Distribution
Table 5
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
t(2)
20
0.5068
0.4482
0.5138
0.5632
0.5282
0.2806
0.3774
0.5268
30
0.6834
0.5832
0.6552
0.7016
0.6908
0.3946
0.4228
0.7004
50
0.8538
0.7782
0.8370
0.8812
0.8572
0.5640
0.4800
0.8726
80
0.9602
0.9200
0.9554
0.9646
0.9566
0.8010
0.5420
0.9658
t(4)
20
0.2270
0.1768
0.2114
0.2898
0.2410
0.0922
0.1698
0.2450
30
0.3002
0.2182
0.2764
0.3788
0.3338
0.1084
0.2164
0.3398
50
0.4150
0.3176
0.3794
0.5400
0.4520
0.1388
0.2468
0.4784
80
0.5558
0.3994
0.5210
0.7064
0.6282
0.2094
0.2784
0.6760
t(7)
20
0.1162
0.0952
0.1006
0.1670
0.1398
0.0492
0.1066
0.1346
30
0.1404
0.1008
0.1306
0.2222
0.1806
0.0552
0.1188
0.1664
50
0.1806
0.1272
0.1578
0.2954
0.2362
0.0502
0.1422
0.2276
80
0.2380
0.1618
0.2086
0.4010
0.3122
0.0650
0.1590
0.3324
Cauchy(0,1)
20
0.8780
0.8386
0.8898
0.8622
0.8674
0.7012
0.6368
0.8450
30
0.9672
0.9410
0.9622
0.9574
0.9610
0.8606
0.6910
0.9542
50
0.9976
0.9950
0.9964
0.9954
0.9958
0.9712
0.7424
0.9976
80
1.0000
1.0000
0.9998
0.9998
0.9998
0.9992
0.8882
1.0000
Cauchy(0,5)
20
0.8778
0.8374
0.8796
0.8650
0.8704
0.6902
0.6454
0.8550
30
0.9628
0.9414
0.9648
0.9512
0.9590
0.8664
0.6950
0.9542
50
0.9968
0.9948
0.9976
0.9968
0.9966
0.9730
0.7468
0.9962
80
1.0000
1.0000
1.0000
0.9998
1.0000
0.9996
0.8872
1.0000
Logistic
20
0.1090
0.0872
0.0982
0.1460
0.1138
0.0436
0.0944
0.1158
30
0.1176
0.0908
0.1220
0.1982
0.1474
0.0452
0.1044
0.1482
50
0.1562
0.1184
0.1456
0.2620
0.1986
0.0414
0.1216
0.1900
80
0.2098
0.1406
0.1870
0.3474
0.2662
0.0468
0.1266
0.2908
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 .
