Goodness-of-fit testing for the inverse Gaussian distribution based on new entropy estimation using ranked set sampling and double ranked set sampling

Environmental Systems Research

Table 5 Monte Carlo RMSEs and bias values of the entropy estimators VE_{(
m,n
)} and AE_{(
m,n
)} for the standard normal distribution and H(f) = 1.419 using DRSS

n	m	VE _(m,n)		AE _(m,n)
		Bias	RMSE	Bias	RMSE
10	1	−0.415021	0.472162	−0.352434	0.416211
	2	−0.373395	0.412666	−0.262149	0.316029
	3	−0.427401	0.459119	−0.254450	0.303820
	4	−0.492911	0.518275	−0.264683	0.310442
	5	−0.554351	0.577281	−0.267798	0.312339
20	1	−0.350703	0.383160	−0.323780	0.359592
	2	−0.245907	0.277809	−0.190733	0.231106
	3	−0.246496	0.276941	−0.158832	0.201924
	4	−0.262789	0.290545	−0.148107	0.194728
	5	−0.291340	0.317967	−0.145734	0.191755
	6	−0.316105	0.341597	−0.147800	0.195946
	7	−0.349246	0.373132	−0.150312	0.199934
	8	−0.384526	0.406764	−0.152801	0.203493
	9	−0.416151	0.436696	−0.156902	0.205954
	10	−0.445901	0.465518	0.159050	0.207883
30	1	−0.321940	0.345223	−0.307781	0.332609
	2	−0.206709	0.231560	−0.169564	0.198438
	3	−0.187163	0.212774	−0.129694	0.163913
	4	−0.190073	0.215577	−0.114103	0.152713
	5	−0.199843	0.224569	−0.103570	0.145964
	6	−0.214636	0.239021	−0.100510	0.146417
	7	−0.231613	0.255278	−0.095517	0.143483
	8	−0.247340	0.271084	−0.094560	0.145579
	9	−0.268298	0.291044	−0.091548	0.145394
	10	−0.286538	0.308661	−0.094236	0.149024
	11	−0.305310	0.326485	−0.093843	0.150300
	12	−0.324892	0.346062	−0.096171	0.152896
	13	−0.343097	0.363236	−0.096892	0.153854
	14	−0.369990	0.388586	−0.100541	0.155029
	15	−0.387740	0.406081	−0.101202	0.156143

Critical points at significance level 0.05 of the test statistic are given in Table 6. The optimal choice of the window size for a given sample size in the estimation of entropy using spacing's is still open problem for testing goodness-of-fit. The bold fonts in Table 6 are the largest critical values based on SRS, RSS and DRSS. For the suggested test, the optimal window size values are summarized in Table 7.