An Empirical Likelihood Ratio-Based Omnibus Test for Normality with an Adjustment for Symmetric Alternatives

Table 4

Type I error rates along with the standard error for the proposed test statistics using .

	Simulated probabilities of the type I error along with standard error
	Alpha	Sample size	CUSUM-type test statistic	Shiryaev–Roberts test statistic
	0.01	20	0.009434  1.3665e − 04	0.009415  1.3655e − 04
		50	0.010655  1.4515e − 04	0.009971  1.4047e − 04
		100	0.009746  1.3890e − 04	0.009969  1.4046e − 04
	0.05	20	0.049903  3.0701e − 04	0.049897  3.0788e −04
		50	0.051580  3.1278e − 04	0.049959  3.0808e − 04
		100	0.048599  3.0406e − 04	0.051496  3.1254e − 04
	0.10	20	0.102288  4.2854e − 04	0.099721  4.2373e − 04
		50	0.102146  4.2825e −04	0.099881  4.2418e − 004
		100	0.096571  4.1769e − 04	0.102983  4.2982e − 04
	0.01	20	0.009358  1.3612e − 04	0.009790  1.3921e − 04
		50	0.010421  1.4357e − 04	0.009792  1.3920e − 04
		100	0.009779  1.3912e − 04	0.009636  1.3812e − 04
	0.05	20	0.050542  3.0979e − 04	0.049276  3.0608e − 04
		50	0.050920  3.1087e − 04	0.048651  3.0424e − 04
		100	0.048685  3.0434e − 04	0.051655  3.1296e − 04
	0.10	20	0.102156  4.2829e − 04	0.099039  4.2243e − 04
		50	0.103031  4.2990e − 04	0.098468  4.2135e − 04
		100	0.095373  4.1537e − 04	0.101721  4.2747e − 04
	0.01	20	0.009227  1.3520e − 04	0.009651  1.3824e − 04
		50	0.010575  1.4462e − 04	0.010104  1.4138e − 04
		100	0.009551  1.3751e − 04	0.009495  1.3713e − 04
	0.05	20	0.050768  3.1043e − 04	0.048988  3.0522e − 04
		50	0.050882  3.1077e − 04	0.048730  3.0448e − 04
		100	0.049044  3.0540e − 04	0.050758  3.1041e − 04
	0.10	20	0.101926  4.2786e − 04	0.100764  4.2569e − 04
		50	0.102209  4.2838e − 04	0.099809  4.2387e − 04
		100	0.096280  4.1714e − 04	0.101345  4.2678e − 04
	0.01	20	0.009416  1.3648e − 04	0.009651  1.3824e − 04
		50	0.010531  1.4432e − 04	0.009910  1.4007e − 04
		100	0.010028  1.4086e − 04	0.009329  1.3592e − 04
	0.05	20	0.049686  3.0729e − 04	0.049465  3.0664e − 04
		50	0.051571  3.1275e − 04	0.049887  3.0788e − 04
		100	0.048452  3.0363e − 04	0.050863  3.1072e − 04
	0.10	20	0.101373  4.2683e − 04	0.099383  4.2307e − 04
		50	0.101062  4.2625e − 04	0.099168  4.2267e − 04
		100	0.095026  4.1470e − 04	0.102562  4.2904e − 04
	0.01	20	0.009383  1.3621e − 04	0.009415  1.3639e − 04
		50	0.010662  1.4521e − 04	0.010074  1.4119e − 04
		100	0.009592  1.3782e − 04	0.009724  1.3874e − 04
	0.05	20	0.049308  3.0618e − 04	0.049587  3.0699e − 04
		50	0.051753  3.1325e − 04	0.049020  3.0533e − 04
		100	0.048467  3.0368e − 04	0.050790  3.1050e − 04
	0.10	20	0.101744  4.2752e − 04	0.099688  4.2366e − 04
		50	0.100745  4.2563e − 04	0.099476  4.2326e − 04
		100	0.096317  4.1722e − 04	0.102060  4.2809e − 04

Note. Monte Carlo simulations were conducted using 500,000 replications.