A Simple Empirical Likelihood Ratio Test for Normality Based on the Moment Constraints of a Half-Normal Distribution
Table 8
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
20
0.6446
0.5692
0.6440
0.6556
0.6498
0.4612
0.5688
0.5542
30
0.8060
0.7178
0.7934
0.8080
0.8072
0.6210
0.6678
0.7242
50
0.9492
0.8900
0.9394
0.9414
0.9410
0.7820
0.7782
0.8872
80
0.9928
0.9762
0.9892
0.9924
0.9924
0.9294
0.8410
0.9726
20
0.3180
0.2368
0.2848
0.3606
0.3142
0.1638
0.2790
0.2744
30
0.4086
0.3246
0.3884
0.4810
0.4518
0.2262
0.3584
0.3718
50
0.5912
0.4618
0.5538
0.6592
0.6360
0.3202
0.4830
0.5344
80
0.7642
0.6370
0.7296
0.8290
0.8108
0.4626
0.5826
0.7084
20
0.1490
0.1138
0.1404
0.1934
0.1692
0.0766
0.1420
0.1492
30
0.1958
0.1424
0.1736
0.2722
0.2318
0.0940
0.1876
0.1936
50
0.2846
0.1968
0.2522
0.3834
0.3372
0.1194
0.2556
0.2738
80
0.3848
0.2756
0.3542
0.5102
0.4556
0.1620
0.3300
0.3798
SN(0,1,2)
20
0.0896
0.0756
0.0882
0.1054
0.1068
0.0636
0.0978
0.0710
30
0.1194
0.0912
0.1062
0.1214
0.1422
0.0784
0.1336
0.0792
50
0.1666
0.1260
0.1488
0.1810
0.1968
0.1164
0.2080
0.0800
80
0.2434
0.1918
0.2258
0.2712
0.2940
0.1550
0.3074
0.0890
SN(0,1,5)
20
0.2406
0.1744
0.2212
0.2060
0.2660
0.2092
0.2810
0.1144
30
0.3586
0.2604
0.3152
0.3056
0.4230
0.3346
0.4742
0.1278
50
0.5796
0.4098
0.5430
0.4768
0.6672
0.5250
0.7110
0.1372
80
0.8080
0.6084
0.7554
0.7378
0.8888
0.7394
0.9020
0.1722
SC(0,2,5)
20
0.9660
0.9360
0.9658
0.9410
0.9736
0.9436
0.9462
0.8482
30
0.9978
0.9884
0.9954
0.9910
0.9970
0.9882
0.9774
0.9524
50
1.0000
0.9998
1.0000
0.9998
0.9998
0.9992
0.9844
0.9940
80
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.9976
0.9998
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 .
