Normal Poisson Simulation Study

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 51 Table 4.5: The empirical mean square errors MSE 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 with µ j = 0 .5, k ∈ {2,4,6,8}, α = X j and β = k. k n MSET n;k ,p,t MSEU n;k ,p,t MSEB n;k ,p,t,X j ,k 2 3 0.3148 - 0.1206 10 0.0600 0.0491 0.0415 20 0.0278 0.0251 0.0229 30 0.0191 0.0177 0.0166 60 0.0083 0.0079 0.0076 100 0.0051 0.0050 0.0049 300 0.0017 0.0017 0.0017 500 0.0010 0.0010 0.0010 1000 0.0005 0.0005 0.0005 4 5 0.7724 - 0.0280 10 0.1085 0.0306 0.0141 20 0.0259 0.0120 0.0082 30 0.0087 0.0048 0.0038 60 0.0033 0.0024 0.0021 100 0.0017 0.0014 0.0013 300 0.0004 0.0004 0.0004 500 0.0003 0.0002 0.0002 1000 0.0001 0.0001 0.0001 6 7 1.6371 - 0.0046 10 0.1646 0.0074 0.0017 20 0.0085 0.0012 0.0006 30 0.0033 0.0007 0.0005 60 0.0009 0.0004 0.0003 100 0.0004 0.0002 0.0002 300 0.0001 0.0001 0.0001 500 0.0000 0.0000 0.0000 1000 0.0000 0.0000 0.0000 8 10 0.1762 - 0.0001 20 0.0210 0.0008 0.0002 30 0.0016 0.0001 0.0001 60 0.0002 0.0000 0.0000 100 0.0001 0.0000 0.0000 300 0.0000 0.0000 0.0000 500 0.0000 0.0000 0.0000 1000 0.0000 0.0000 0.0000 when the mean of the Poisson component becomes smaller. For normal- Poisson distribution with µ j = 0 .5, we have many zero values in the samples. However, this situation does not affect the generalized variance estimation as we can see that T n;k ,p,t , U n;k ,p,t and B n;k ,p,t,α,β have the same behavior for all values of µ j . The MSE in Table 4.5, 4.7 and 4.9 are displayed as graphs presented in Figure 4.4, Figure 4.5 and Figure 4.6. From those figures we conclude that 52 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=2 and mu_j=0.5 Sample Size Mean Square Error 0.00 0.01 0.02 0.03 0.04 0.05 a k=2 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=4 and mu_j=0.5 Sample Size Mean Square Error 0.00 0.02 0.04 0.06 0.08 0.10 b k=4 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=6 and mu_j=0.5 Sample Size Mean Square Error 0.00 0.05 0.10 0.15 c k=6 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=8 and mu_j=0.5 Sample Size Mean Square Error 0.00 0.05 0.10 0.15 d k=8 Figure 4.4: Bargraphs of the mean square errors of T n;k ,p,t , U n;k ,p,t and B n;k ,p,t,α,β for normal-Poisson with µ j = 0 .5 and k ∈ {2,4,6,8}. the U n;k ,p,t is preferable than T n;k ,p,t because it always has smaller MSE val- ues when sample sizes are small; i.e. n 6 30. In this situation, the difference between U n;k ,p,t and T n;k ,p,t increases when the dimension k increases. The lack of U n;k ,p,t when n kX j can be solved by using Bayesian estimator as an alternative for obtaining a better estimation than ML estimator. How- ever, UMVU is the only unbiased estimator while ML and Bayesian are asymptotically unbiased. Notice that for µ j = 5 the MSE values of Bayesian estimator for n ≥ 30 are greater than the ML estimator. This kind of behavior also happens for µ j = 10 we do not present the result here. Hence we conclude that ML estimator is better than the Bayesian estimator for large values of µ j . 53 Table 4.6: 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 = 1 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 1 3 1.3711 1.4982 1.0349 1.3130 0.8775 0.9589 10 1.0810 0.6589 0.9817 0.6286 0.9083 0.5536 20 1.0424 0.4471 0.9925 0.4363 0.9498 0.4074 30 1.0329 0.3817 0.9996 0.3756 0.9694 0.3583 60 1.0184 0.2661 1.0017 0.2639 0.9858 0.2576 100 1.0066 0.2016 0.9966 0.2006 0.9870 0.1977 300 1.0112 0.1153 1.0079 0.1151 1.0045 0.1146 500 0.9986 0.0942 0.9966 0.0941 0.9946 0.0938 1000 0.9998 0.0641 0.9988 0.0641 0.9978 0.0640 4 1 5 2.6283 5.0058 1.0721 2.7753 0.5192 0.9888 10 1.7362 2.2949 1.0422 1.6267 0.6617 0.8746 20 1.3276 1.1713 1.0073 0.9588 0.7782 0.6866 30 1.2274 0.8892 1.0167 0.7750 0.8482 0.6145 60 1.1111 0.5643 1.0085 0.5250 0.9170 0.4657 100 1.0647 0.4448 1.0038 0.4260 0.9471 0.3957 300 1.0245 0.2389 1.0043 0.2354 0.9846 0.2296 500 1.0092 0.1889 0.9972 0.1872 0.9854 0.1845 1000 1.0013 0.1272 0.9953 0.1267 0.9894 0.1257 6 1 7 4.5153 12.8404 0.9378 4.0255 0.2452 0.6974 10 3.6865 8.1473 1.1642 3.3992 0.3893 0.8603 20 1.9674 2.9034 1.0227 1.7467 0.5462 0.8061 30 1.5605 1.8825 0.9901 1.3133 0.6362 0.7675 60 1.2954 1.0360 1.0220 0.8541 0.8075 0.6458 100 1.2084 0.7824 1.0462 0.6957 0.9043 0.5855 300 1.0621 0.3793 1.0109 0.3641 0.9621 0.3436 500 1.0294 0.2778 0.9992 0.2710 0.9699 0.2617 1000 1.0185 0.1939 1.0034 0.1915 0.9885 0.1882 8 1 10 7.4252 31.4171 0.9507 7.0716 0.1444 0.6111 20 2.9347 6.6901 0.8934 2.6109 0.2938 0.6698 30 2.1114 4.0742 0.9266 2.1005 0.4142 0.7992 60 1.5445 1.8064 0.9988 1.2576 0.6477 0.7575 100 1.2712 1.0323 0.9704 0.8197 0.7437 0.6039 300 1.0814 0.5222 0.9863 0.4820 0.8998 0.4345 500 1.0569 0.3933 0.9998 0.3748 0.9458 0.3520 1000 1.0290 0.2634 1.0007 0.2571 0.9732 0.2491 54 Table 4.7: The empirical mean square errors MSE 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 with µ j = 1, k ∈ {2,4,6,8}, α = X j and β = k. k n MSET n;k ,p,t MSEU n;k ,p,t MSEB n;k ,p,t,X j ,k 2 3 2.3824 1.7252 0.9344 10 0.4407 0.3955 0.3149 20 0.2017 0.1904 0.1685 30 0.1468 0.1411 0.1293 60 0.0711 0.0697 0.0665 100 0.0407 0.0403 0.0392 300 0.0134 0.0133 0.0131 500 0.0089 0.0089 0.0088 1000 0.0041 0.0041 0.0041 4 5 27.7093 7.7075 1.2089 10 5.8085 2.6480 0.8794 20 1.4793 0.9193 0.5206 30 0.8424 0.6008 0.4007 60 0.3308 0.2757 0.2238 100 0.2021 0.1815 0.1594 300 0.0577 0.0554 0.0529 500 0.0358 0.0351 0.0342 1000 0.0162 0.0161 0.0159 6 7 177.2319 16.2084 1.0560 10 73.5952 11.5816 1.1131 20 9.3656 3.0514 0.8557 30 3.8580 1.7250 0.7214 60 1.1606 0.7300 0.4541 100 0.6556 0.4861 0.3520 300 0.1477 0.1327 0.1195 500 0.0780 0.0734 0.0694 1000 0.0379 0.0367 0.0356 8 10 1028.3183 50.0103 1.1055 20 48.5011 6.8281 0.9473 30 17.8342 4.4177 0.9819 60 3.5595 1.5815 0.6979 100 1.1392 0.6728 0.4304 300 0.2793 0.2325 0.1988 500 0.1579 0.1405 0.1268 1000 0.0702 0.0661 0.0628 55 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=2 and mu_j=1 Sample Size Mean Square Error 0.0 0.1 0.2 0.3 0.4 a k=2 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=4 and mu_j=1 Sample Size Mean Square Error 1 2 3 4 5 b k=4 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=6 and mu_j=1 Sample Size Mean Square Error 10 20 30 40 50 60 70 c k=6 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=8 and mu_j=1 Sample Size Mean Square Error 200 400 600 800 1000 d k=8 Figure 4.5: Bargraphs of the mean square errors of T n;k ,p,t , U n;k ,p,t and B n;k ,p,t,α,β for normal-Poisson with µ j = 1 and k ∈ {2,4,6,8}. 56 Table 4.8: 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 normal-Poisson from 1000 replications for given target value µ k j = 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 25 3 26.59 13.49 24.93 13.06 17.02 8.63 10 25.50 7.04 25.00 6.97 21.42 5.92 20 25.30 5.13 25.05 5.10 23.06 4.67 30 25.14 4.07 24.97 4.06 23.59 3.82 60 25.10 2.84 25.01 2.83 24.29 2.75 100 25.14 2.25 25.09 2.25 24.65 2.21 300 25.02 1.28 25.01 1.28 24.86 1.27 500 25.01 1.01 25.00 1.01 24.91 1.01 1000 25.00 0.68 25.00 0.68 24.95 0.68 4 625 10 722.35 379.32 643.63 348.05 275.30 144.56 20 656.31 267.59 618.66 256.15 384.72 156.86 30 643.63 213.01 618.63 206.83 444.80 147.21 60 640.46 148.59 627.87 146.40 528.56 122.63 100 635.85 113.86 628.30 112.85 565.60 101.28 300 627.40 64.85 624.90 64.65 603.00 62.33 500 624.47 49.86 622.97 49.77 609.73 48.68 1000 623.99 36.45 623.24 36.42 616.56 36.02 6 15625 10 21464.28 21253.41 16234.02 17018.96 2266.48 2244.22 20 18437.60 11153.66 15946.65 9905.34 5118.94 3096.66 30 17479.01 8777.20 15851.79 8099.08 7126.43 3578.58 60 16272.58 5688.96 15485.73 5460.96 10143.13 3546.08 100 16036.24 4356.63 15564.94 4250.16 12000.40 3260.19 300 15724.19 2492.89 15567.97 2472.28 14244.22 2258.26 500 15670.97 1874.69 15577.30 1865.36 14764.51 1766.25 1000 15708.09 1336.94 15661.09 1333.60 15245.44 1297.56 8 390625 10 695388.58 1228177.84 420480.34 830043.27 13526.61 23890.36 20 536931.48 478032.35 411437.57 381388.82 53753.78 47857.21 30 451337.25 314617.29 376214.48 268982.30 88537.24 61717.36 60 423826.51 195474.33 386517.24 180478.34 177727.86 81970.41 100 419253.53 153381.74 396662.57 146155.19 245277.48 89733.50 300 394950.63 83679.72 387661.84 82331.65 328601.12 69621.99 500 392989.13 63923.39 388617.32 63302.46 351700.23 57207.36 1000 390946.84 43047.19 388764.23 42836.90 369748.20 40713.01 57 Table 4.9: The empirical mean square errors MSE 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 with µ j = 5, k ∈ {2,4,6,8}, α = X j and β = k. k n MSET n;k ,p,t MSEU n;k ,p,t MSEB n;k ,p,t,X j ,k 2 3 1.8452E+02 1.7062E+02 1.3821E+02 10 4.9851E+01 4.8634E+01 4.7817E+01 20 2.6385E+01 2.6036E+01 2.5606E+01 30 1.6613E+01 1.6486E+01 1.6607E+01 60 8.0631E+00 8.0272E+00 8.0455E+00 100 5.0817E+00 5.0602E+00 4.9898E+00 300 1.6345E+00 1.6329E+00 1.6328E+00 500 1.0213E+00 1.0208E+00 1.0211E+00 1000 4.6392E-01 4.6384E-01 4.6441E-01 4 10 1.5336E+05 1.2149E+05 1.4319E+05 20 7.2587E+04 6.5653E+04 8.2341E+04 30 4.5719E+04 4.2820E+04 5.4142E+04 60 2.2319E+04 2.1442E+04 2.4339E+04 100 1.3081E+04 1.2745E+04 1.3786E+04 300 4.2109E+03 4.1801E+03 4.3684E+03 500 2.4859E+03 2.4808E+03 2.6029E+03 1000 1.3299E+03 1.3296E+03 1.3687E+03 6 10 4.8580E+08 2.9002E+08 1.8349E+08 20 1.3231E+08 9.8219E+07 1.1997E+08 30 8.0477E+07 6.5646E+07 8.5032E+07 60 3.2784E+07 2.9841E+07 4.2626E+07 100 1.9149E+07 1.8068E+07 2.3767E+07 300 6.2243E+06 6.1154E+06 7.0063E+06 500 3.5166E+06 3.4819E+06 3.8601E+06 1000 1.7943E+06 1.7798E+06 1.8277E+06 8 10 1.6013E+12 6.8986E+11 1.4277E+11 20 2.4992E+11 1.4589E+11 1.1577E+11 30 1.0267E+11 7.2559E+10 9.5066E+10 60 3.9313E+10 3.2589E+10 5.2044E+10 100 2.4346E+10 2.1398E+10 2.9178E+10 300 7.0210E+09 6.7873E+09 8.6942E+09 500 4.0918E+09 4.0112E+09 4.7878E+09 1000 1.8532E+09 1.8385E+09 2.0934E+09 58 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=2 and mu_j=5 Sample Size Mean Square Error 10 20 30 40 a k=2 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=4 and mu_j=5 Sample Size Mean Square Error 20000 40000 60000 80000 100000 140000 b k=4 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=6 and mu_j=5 Sample Size Mean Square Error 0e+00 1e+08 2e+08 3e+08 4e+08 c k=6 10 20 30 60 100 300 500 1000 ML UMVU BAYES MSE bargraph for k=8 and mu_j=5 Sample Size Mean Square Error 0.0e+00 5.0e+11 1.0e+12 1.5e+12 d k=8 Figure 4.6: Bargraphs of the mean square errors of T n;k ,p,t , U n;k ,p,t and B n;k ,p,t,α,β for normal-Poisson with µ j = 5 and k ∈ {2,4,6,8}.

5. CHARACTERIZATIONS OF NORMAL POISSON

MODELS A normal-Poisson model is a special case of normal stable Tweedie NST models. Similar to all NST models, this model was introduced by Boubacar Maïnassara and Kokonendji for particular case of j, i.e. j = 1. They de- fined that a k-variate normal-Poisson model is composed by distribution of random vector X = X 1 , . . . , X k ⊤ where X 1 is a univariate non-negative Pois- son variable and X 2 , . . . , . . . , X k ⊤ =: X c 1 given X 1 are k − 1 real independent Gaussian variables with variance X 1 . 5.1. Definition and Properties Following Boubacar Maïnassara and Kokonendji 2014 with j = 1 we here describe the family of multivariate normal-Poisson j models. Definition 5.1.1. Let j ∈ {1,2,...,k} and k 1. For a k-dimensional normal-Poisson random vector X, it must hold two conditions: 1. X j is a univariate Poisson random variable 2. X 1 , . . . , X j− , X j+1 . . . , X k ⊤ given X j =: X c j |X j = x j , follows the k−1-variate normal N k−1 , X j I k−1 distribution, where I ℓ = diag ℓ 1 , . . . , 1 denotes the ℓ × ℓ unit matrix. In order to satisfy the second condition we need X j 0. But in practice it is possible to have X j = 0 in the Poisson component. In this case, the corresponding normal components are degenerated as the Dirac mass δ which makes their values become 0s. In the previous chapter it has been shown that zero values in X j do not affect the estimation of the generalized variance of normal-Poisson. Theoretically, the problem may happens when we have many zero values in the sample; that is the distribution of normal components will change into Dirac distribution. Then by Definition 5.1.1 and Equation 3.4, for a fixed power of convo- lution t 0 and given j ∈ {1,2,...,k}, denote F t;j = F ν 1 ,t;j with ν 1 ,t;j := ν ∗t 1 ,j . The NEF Kotz et al., 2000, Chapter 54 of a k-dimensional normal-Poisson j random vector X is generated by ν 1 ,t;j dx = t x j x j −1 2 πx j k−12 exp   −t− 1 2x j X ℓ,j x 2 ℓ   1 x j ∈N\{0} δ x j dx j Y ℓ,j dx ℓ , 5.1 where I A is the indicator function of the set A. Since t 0 then ν 1 ,t;j is known to be an infinitely divisible measure; see, e.g., Sato 1999. For simplicity, henceforth we replace ν 1 ,t;j by ν t;j . 59 60 The cumulant function which is the logarithm of the Laplace transform of ν t;j is: K ν t;j θ = log Z R k expθ ⊤ x ν t;j dx = log Z R exp θ j x j    Y ℓ,j Z R exp θ ℓ x ℓ ν N0,x j dx ℓ   ν Pt dx j = log Z R exp θ j x j    Y ℓ,j exp x j θ 2 ℓ 2   ν Pt dx j = t exp   θ j + 1

2 X

ℓ,j θ 2 ℓ    5.2 where ν N0,x j = 1 2 πx j 1 2 exp − 1 2x j x 2 ℓ and ν Pt = t x j x j −1 exp−t1 x j ∈N\{0} δ x j . The function K ν t;j θ is finite for all θ in the canonical domain Θ ν t;j =   θ ∈ R k ; θ ⊤ ˜θ c j := θ j + 1

2 X

ℓ,j θ 2 ℓ    5.3 with θ = θ 1 , . . . , θ k ⊤ and ˜θ c j := θ 1 , . . . , θ j−1 , θ j = 1 , θ j+1 , . . . , θ k ⊤ . 5.4 The probability distribution of normal-Poisson j which is a member of NEF is given by Pθ; ν t;j dx = exp{θ ⊤ x − K ν t;j θ}ν t;j dx . From 5.2 we can calculate the first derivative of the cumulant function that produces a k-vector as the mean vector of F t;j , and also its second derivative which is a k × k matrix that represents the covariance matrix. The first derivative of the cumulant function with respect to θ is obtained as follow: 61 K ′ ν t;j θ = ∂ ∂θ s K ν t;j θ s=1 ,...,k =                 t θ 1 exp   θ j + 1 2 P ℓ,j θ 2 ℓ    ... t θ j−1 exp   θ j + 1 2 P ℓ,j θ 2 ℓ    t exp   θ j + 1 2 P ℓ,j θ 2 ℓ    t θ j+1 exp   θ j + 1 2 P ℓ,j θ 2 ℓ    ... t θ k exp   θ j + 1 2 P ℓ,j θ 2 ℓ                    = tθ exp   θ j + 1

2 X

ℓ,j θ 2 ℓ            θ 1 ... θ j−1 1 θ j+1 ... θ k         . = t exp   θ j + 1

2 X

ℓ,j θ 2 ℓ   θθ c j = K ν t;j θ × ˜θ c j Then the second derivative of the cumulant function with respect to θ is obtained as follow: