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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

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

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
  1. 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.