Marshall–Olkin Extended Gumbel Type-II Distribution: Properties and Applications
Table 4
MLE and their sample statistics for some simulated samples and true values = 2, = 4, = 6, and = 5.
Nelder–Mead
Quasi-Newton
Fletcher–Reeves
n
Parameters
Min
Max
Mean
Std
Min
Max
Mean
Std
Min
Max
Mean
Std
50
0.154
1471.2
4.234
39.642
0.268
11.623
1.813
0.668
0.504
7.319
1.74
0.448
0.456
22.541
5.175
2.448
0.938
12.668
4.213
0.818
2.018
9.891
4.211
0.44
1.815
34.877
5.773
1.590
3.525
11.051
6.099
0.62
3.794
8.582
6.262
0.561
3672.7
2083.4
3.165
88.678
1.719
11.4
4.865
0.687
4.235
11.538
4.664
0.664
Iteration
79
501
141
38
16
295
46
25
4
922
835
41
100
0.293
430.19
1.963
7.577
0.194
8.17
1.77
0.479
0.595
4.014
1.771
0.335
0.526
21.514
4.998
1.614
1.336
11.559
4.203
0.713
2.079
7.312
4.169
0.317
2.186
32.628
5.594
0.879
3.702
8.454
6.026
0.443
4.291
8.121
6.22
0.423
854.3
36.5
5.208
13.651
1.443
8.815
4.848
0.594
1.423
7.764
4.725
0.517
Iteration
79
501
132
27
19
949
50
39
7
921
842
31
300
0.565
5.146
1.625
0.505
0.519
13.851
1.81
0.361
0.659
3.602
1.80
0.301
2.405
9.443
4.908
0.896
2.187
12.288
4.109
0.411
2.612
7.402
4.163
0.265
4.187
7.439
5.501
0.421
3.36
7.595
6.049
0.335
4.558
7.328
6.208
0.334
6.865
13.148
5.534
0.877
0.151
11.046
4.836
0.426
0.838
8.322
4.757
0.424
Iteration
79
273
127
22
2
1308
51
58
8
924
858
31
500
0.73
3.868
1.596
0.378
0.411
4.372
1.805
0.309
0.494
4.341
1.892
0.261
2.969
7.92
4.899
0.696
1.9
8.282
4.106
0.380
1.945
7.902
4.105
0.236
4.442
6.689
5.481
0.325
4.181
7.718
6.039
0.323
3.927
7.240
6.024
0.291
3.78
17.343
5.553
0.715
1.561
9.143
4.82
0.396
0.928
6.997
4.724
0.387
Iteration
85
255
127
25
20
934
52
61
14
990
866
30
1000
0.893
2.781
1.559
0.261
0.473
4.319
1.779
0.259
0.661
4.016
1.824
0.256
3.371
6.974
4.861
0.50
2.099
8.383
4.099
0.376
2.569
7.690
4.141
0.228
4.721
6.551
5.476
0.233
4.256
7.148
5.984
0.217
4.333
7.307
6.182
0.215
2.964
13.442
5.566
0.604
0.759
8.398
4.78
0.390
0.325
6.705
4.788
0.385
Iteration
83
253
128
23
2
1349
54
64
12
950
867
29
From Tables 3 and 4, it can be seen that with increasing sample size, the standard deviations for the estimates tend to decrease and the Fletcher–Reeves method provides a better estimate as compared with the Nelder–Mead method and Quasi-Newton method with increasing sample size.
