Normal inverse-Gaussian Simulation Study

48 Table 4.3: The expected values with standard errors and MSE of T n;k ,p,t and U n;k ,p,t for normal inverse-Gaussian with 1000 replications for given target value µ k+2 j = 1 and k ∈ {2,4,6,8}. k Target n E T n;k ,p,t StdT n;k ,p,t E U n;k ,p,t StdU n;k ,p,t MSE T n;k ,p,t MSE U n;k ,p,t 2 1 3 2.0068 4.9227 0.9135 0.8235 25.2469 0.6856 10 1.4249 2.8513 1.0316 0.4388 8.3103 0.1935 20 1.5936 1.8951 1.1340 0.3718 3.9439 0.1562 30 1.3677 1.0155 1.1641 0.2668 1.1664 0.0981 60 1.0846 0.5341 1.1104 0.1856 0.2924 0.0466 100 1.0819 0.5166 1.1102 0.1675 0.2735 0.0402 300 1.0006 0.2570 1.0843 0.0919 0.0660 0.0156 500 1.0356 0.1890 1.1374 0.0727 0.0370 0.0242 1000 1.0156 0.1219 1.0116 0.0670 0.0151 0.0115 4 1 5 9.3836 30.0947 1.3196 1.1323 975.9726 1.3843 10 4.6547 13.8643 1.2837 0.8153 205.5754 0.7452 20 2.7487 5.1845 1.2963 0.6189 29.9373 0.4709 30 1.4822 2.1166 1.1854 0.4572 4.7125 0.2434 60 1.3095 1.1051 1.2560 0.3054 1.3170 0.1588 100 1.1673 0.8467 1.2264 0.2671 0.7449 0.1226 300 1.0849 0.4296 1.2542 0.1520 0.1918 0.0877 500 1.0350 0.2839 1.0762 0.0914 0.0818 0.0416 1000 1.0107 0.2080 1.0102 0.1137 0.0434 0.0337 6 1 7 20.4865 113.4633 0.9423 0.9984 13253.6508 1.0001 10 12.1032 55.7841 1.0596 0.8610 3235.1488 0.7449 20 3.4498 10.3056 1.0054 0.5933 112.2060 0.3520 30 2.1422 3.2262 1.0246 0.4970 11.7130 0.2476 60 1.8236 2.6064 1.0587 0.3744 7.4717 0.1436 100 1.2468 1.1599 1.0129 0.2643 1.4062 0.1170 300 1.0781 0.4953 1.0568 0.1596 0.2514 0.0929 500 1.0815 0.4065 1.0230 0.1110 0.1719 0.0922 1000 1.0207 0.2816 1.0204 0.0775 0.0798 0.0760 8 1 10 27.9651 106.4417 1.1645 1.2832 12056.9414 1.6737 20 10.2639 47.3683 1.2127 0.9227 2329.5787 0.8674 30 5.8903 14.2638 1.2634 0.8024 227.3707 0.7133 60 1.8667 3.2137 1.1402 0.4894 11.0792 0.2504 100 1.5251 1.8103 1.1340 0.3734 3.5530 0.1591 300 1.2059 0.8122 1.1398 0.2275 0.7021 0.1571 500 1.1817 0.6075 1.1210 0.3032 0.4021 0.1200 1000 1.0325 0.3189 1.1125 0.0564 0.1027 0.0910

4.2.3. Normal Poisson

Again by fixing j = 1 we set several sample sizes n varied from 5 until 1000 and we generated 1000 samples for each n. However, for normal Poisson case, to see the effect of zero values proportion within X j , we also consider small mean values on the Poisson component because PX j = 0 = exp−µ j , then we set µ j = 0 .5, 1 and 5. We also used Theorem 4.1.3 for calculating Bayesian estimator in this simulation, we assume that the parameters of 49 10 20 30 60 100 300 500 1000 ML UMVU MSE bargraph for k=2 and mu_j=1 Sample Size Mean Square Error 2 4 6 8 a k=2 10 20 30 60 100 300 500 1000 ML UMVU MSE bargraph for k=4 and mu_j=1 Sample Size Mean Square Error 50 100 150 200 b k=4 10 20 30 60 100 300 500 1000 ML UMVU MSE bargraph for k=6 and mu_j=1 Sample Size Mean Square Error 500 1000 1500 2000 2500 3000 c k=6 10 20 30 60 100 300 500 1000 ML UMVU MSE bargraph for k=8 and mu_j=1 Sample Size Mean Square Error 2000 4000 6000 8000 10000 12000 d k=8 Figure 4.3: Bargraphs of the mean square errors of T n;k ,p,t and U n;k ,p,t for normal inverse Gaussian with µ j = 1 and k ∈ {2,4,6,8}. prior distribution depend on sample mean of the Poisson component X j and the space dimension k. The results are presented in Tables 4.4 - 4.9. From the expected values and the standard errors in Tables 4.4, 4.6 and 4.8, we can observe the performance of ML T n;k ,p,t , UMVU U n;k ,p,t and Bayesian B n;k ,p,t,α,β estimations on the generalized variance. The values of all estimators converge; but U n;k ,p,t , which is the unbiased estimator, always approximates the target m k j more accurately than T n;k ,p,t and B n;k ,p,t,α,β for small sample sizes n 6 30. Notice that U n;k ,p,t can be calculated only if nX j k; for this reason U n;k ,p,t is not available for some n when m j = 0 .5. In this case, for other n where U n;k ,p,t is available, we can observe that B n;k ,p,t,X j ,k is closer to U n;k ,p,t than T n;k ,p,t . Thus if µ j is small and U n;k ,p,t is not available, we can get a good estimation 50 Table 4.4: The expected values with empirical standard errors of T n;k ,p,t , U n;k ,p,t and B n;k ,p,t,α,β for normal-Poisson from 1000 replications for given target value µ k j = 0 .5 k with k ∈ {2,4,6,8}, α = X j and β = k. k Target n T n;k ,p,t Std T U n;k ,p,t Std U B n;k ,p,t,X j ,k Std B 2 0.25 3 0.3930 0.5426 - 0.2515 0.3473 10 0.2868 0.2421 0.2378 0.2212 0.2410 0.2034 20 0.2652 0.1660 0.2407 0.1583 0.2416 0.1513 30 0.2642 0.1374 0.2476 0.1332 0.2480 0.1290 60 0.2598 0.0903 0.2514 0.0888 0.2515 0.0874 100 0.2534 0.0712 0.2484 0.0705 0.2485 0.0698 300 0.2495 0.0418 0.2478 0.0417 0.2478 0.0415 500 0.2491 0.0313 0.2482 0.0313 0.2482 0.0312 1000 0.2495 0.0221 0.2490 0.0221 0.2490 0.0221 4 0.0625 5 0.2999 0.8462 - 0.0592 0.1672 10 0.1696 0.3115 0.0689 0.1750 0.0646 0.1187 20 0.1089 0.1541 0.0658 0.1097 0.0638 0.0903 30 0.0886 0.0894 0.0617 0.0689 0.0613 0.0618 60 0.0774 0.0559 0.0642 0.0487 0.0639 0.0461 100 0.0704 0.0403 0.0627 0.0370 0.0627 0.0358 300 0.0643 0.0207 0.0618 0.0201 0.0618 0.0199 500 0.0635 0.0158 0.0620 0.0156 0.0620 0.0155 1000 0.0631 0.0115 0.0624 0.0114 0.0624 0.0113 6 0.015625 7 0.2792 1.2521 - 0.0152 0.0680 10 0.1212 0.3918 0.0165 0.0858 0.0128 0.0414 20 0.0427 0.0883 0.0124 0.0345 0.0119 0.0245 30 0.0356 0.0539 0.0151 0.0271 0.0145 0.0220 60 0.0236 0.0281 0.0149 0.0196 0.0147 0.0175 100 0.0211 0.0183 0.0159 0.0145 0.0158 0.0137 300 0.0173 0.0089 0.0157 0.0082 0.0157 0.0081 500 0.0166 0.0068 0.0157 0.0064 0.0157 0.0064 1000 0.0164 0.0044 0.0159 0.0043 0.0159 0.0043 8 0.00390625 10 0.0891 0.4110 - 0.0017 0.0080 20 0.0384 0.1409 0.0054 0.0288 0.0038 0.0141 30 0.0171 0.0383 0.0037 0.0107 0.0033 0.0075 60 0.0081 0.0119 0.0035 0.0058 0.0034 0.0050 100 0.0063 0.0082 0.0038 0.0053 0.0037 0.0048 300 0.0045 0.0031 0.0038 0.0027 0.0038 0.0026 500 0.0045 0.0024 0.0040 0.0022 0.0040 0.0021 1000 0.0041 0.0015 0.0039 0.0014 0.0039 0.0014 by using B n;k ,p,t,X j ,k . While for µ j = 1 and µ j = 5, the Bayesian estimator with prior distribution gammaX j , k produces the closer estimates to the UMVU than ML method. We can improve this Bayesian estimator by using other parameter values of prior distribution. From the MSEs in Tables 4.5,4.7 and 4.9 we can conclude that all estimators are consistent. In this simulation, the proportion of zero values in the samples increases