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}.