A Simple Empirical Likelihood Ratio Test for Normality Based on the Moment Constraints of a Half-Normal Distribution
Table 4
A numerical assessment on power using the Shiryaev-Roberts (S-R) and CUSUM-type (C-t) statistics for the proposed tests ( and ) with increased moment constraints at .
S-R
C-t
S-R
C-t
S-R
C-t
S-R
C-t
(2)
30
0.0416
0.0330
0.0020
0.0010
0.6980
0.6998
0.6166
0.5912
50
0.5142
0.4112
0.1666
0.1356
0.8766
0.8774
0.8320
0.8030
80
0.8732
0.8336
0.7476
0.7184
0.9718
0.9684
0.9544
0.9488
Cauchy(0,1)
30
0.3262
0.3438
0.0000
0.0000
0.9560
0.9556
0.9248
0.9192
50
0.9538
0.9344
0.7246
0.6754
0.9970
0.9974
0.9928
0.9900
80
0.9996
0.9996
0.9964
0.9940
1.0000
1.0000
0.9998
0.9996
Uniform(0,1)
30
0.7230
0.1958
0.7208
0.7206
0.5772
0.5986
0.6996
0.7004
50
0.9458
0.5222
0.9532
0.9474
0.9032
0.9122
0.9434
0.9398
80
0.9966
0.8462
0.9978
0.9980
0.9940
0.9956
0.9986
0.9976
Exp(1)
30
0.0094
0.0304
0.0070
0.0068
0.4638
0.4818
0.3874
0.3772
50
0.0836
0.8096
0.0022
0.0042
0.6274
0.6306
0.5628
0.5380
80
0.3764
0.9972
0.2504
0.2346
0.7942
0.8070
0.7558
0.7506
30
0.0476
0.0136
0.0012
0.0028
0.7168
0.7230
0.6476
0.6280
50
0.5204
0.4172
0.1932
0.1676
0.8904
0.8908
0.8450
0.8294
80
0.8752
0.8610
0.7736
0.7700
0.9714
0.9766
0.9636
0.9558
SN(0,1,5)
30
0.0514
0.0520
0.0486
0.0442
0.1394
0.1242
0.1048
0.0944
50
0.0404
0.0362
0.0350
0.0352
0.1408
0.1432
0.1114
0.0904
80
0.0358
0.0338
0.0272
0.0204
0.1592
0.1646
0.1158
0.1226
Note. Our proposed tests are maximized on , where can take any integer to represent the moment constraints used to maximise the test statistics for specified sample sizes at level of significance using 5,000 simulations. is the sample size. Bold represents the powerful test statistic for the given simulation scenarios.
