A Simple Empirical Likelihood Ratio Test for Normality Based on the Moment Constraints of a Half-Normal Distribution
Table 7
Results of the Monte Carlo power comparisons based on samples with sizes from asymmetric alternative distributions defined on at .
Asymmetric alternative distributions defined on at
Distribution
AD
CVM
JB
SW
DB
SEELR
Exp(1)
20
0.7850
0.5722
0.7222
0.6230
0.8334
0.8384
0.8522
0.3642
30
0.9296
0.7780
0.8922
0.8286
0.9646
0.9754
0.9996
0.4752
50
0.9972
0.9594
0.9878
0.9756
0.9998
0.9992
1.0000
0.6400
80
1.0000
0.9990
0.9998
0.9998
1.0000
1.0000
1.0000
0.8114
Gamma(2,1)
20
0.4590
0.3066
0.4136
0.4080
0.5380
0.4420
0.5684
0.2264
30
0.6662
0.4776
0.6072
0.5852
0.7502
0.6876
0.8094
0.2844
50
0.8960
0.6926
0.8436
0.8242
0.9500
0.9180
0.9668
0.3822
80
0.9840
0.8962
0.9682
0.9782
0.9976
0.9914
0.9984
0.5210
Lognorm(0,1)
20
0.9080
0.7760
0.8846
0.8172
0.9350
0.9210
0.9418
0.6036
30
0.9838
0.9304
0.9730
0.9466
0.9888
0.9906
1.0000
0.7418
50
1.0000
0.9942
0.9998
0.9976
1.0000
1.0000
1.0000
0.9068
80
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9838
Lognorm(0,2)
20
0.9986
0.9904
0.9970
0.9840
0.9990
0.9998
0.9999
0.8894
30
0.9998
0.9998
1.0000
0.9994
1.0000
1.0000
1.0000
0.9684
50
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9988
80
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
Weibull(2,1)
20
0.1348
0.0980
0.1142
0.1258
0.1582
0.1264
0.1626
0.0932
30
0.1828
0.1306
0.1654
0.1704
0.2274
0.1958
0.2718
0.0892
50
0.3050
0.2000
0.2530
0.2738
0.4086
0.3446
0.5202
0.1120
80
0.4954
0.3186
0.4200
0.4346
0.6644
0.5634
0.7812
0.1080
Weibull(0.5,1)
20
0.9962
0.9810
0.9954
0.9562
0.9990
0.9996
0.9986
0.8014
30
1.0000
0.9990
1.0000
0.9972
1.0000
1.0000
1.0000
0.9168
50
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9866
80
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
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 .
