Table 3 MLE estimates for the parameters of \(\mathcal {EW}\) distribution with simulated data from \(\mathcal {EW}\) distribution via BFGS, SANN, and Nelder–Mead algorithms
n | Method | Inference results | b | c | \(\beta\) |
|---|---|---|---|---|---|
50 | BFGS | Estimates | 4.402925 | 4.102611 | 5.949618 |
SE | 5.697425 | 0.751590 | 2.601159 | ||
MSE | 22.795855 | 0.402740 | 9.699106 | ||
SANN | Estimates | 4.449711 | 4.106828 | 5.920183 | |
SE | 6.085270 | 0.789521 | 2.686872 | ||
MSE | 24.098193 | 0.413645 | 8.739624 | ||
Nelder–Mead | Estimates | 4.472318 | 4.107355 | 5.951902 | |
SE | 5.962889 | 0.764355 | 2.603384 | ||
MSE | 25.399142 | 0.420800 | 10.046063 | ||
Time | 0d:8h:50m:45s (31845 s) | ||||
100 | BFGS | Estimates | 4.213534 | 4.113357 | 5.299846 |
SE | 3.566710 | 0.509471 | 1.548855 | ||
MSE | 18.221747 | 0.290249 | 2.756462 | ||
SANN | Estimates | 4.207505 | 4.114013 | 5.297310 | |
SE | 3.650426 | 0.521174 | 1.571037 | ||
MSE | 17.379728 | 0.288609 | 2.753479 | ||
Nelder–Mead | Estimates | 4.249701 | 4.115646 | 5.298766 | |
SE | 3.666593 | 0.513693 | 1.549940 | ||
MSE | 19.982132 | 0.299304 | 2.763964 | ||
Time | 0d:15h:18m:26s (55106 s) | ||||
500 | BFGS | Estimates | 3.155945 | 4.017774 | 5.055031 |
SE | 0.906044 | 0.185317 | 0.627086 | ||
MSE | 0.906766 | 0.035019 | 0.396619 | ||
SANN | Estimates | 3.157440 | 4.017996 | 5.054715 | |
SE | 0.911430 | 0.186253 | 0.629546 | ||
MSE | 0.911503 | 0.035196 | 0.398176 | ||
Nelder–Mead | Estimates | 3.156035 | 4.017759 | 5.055316 | |
SE | 0.906550 | 0.185396 | 0.627336 | ||
MSE | 0.908748 | 0.035088 | 0.397327 | ||
Time | 2d:16h:15m:17s (231317 s) | ||||
1000 | BFGS | Estimates | 3.078456 | 4.009790 | 5.023386 |
SE | 0.609515 | 0.128449 | 0.437921 | ||
MSE | 0.390373 | 0.016572 | 0.189574 | ||
SANN | Estimates | 3.078632 | 4.009775 | 5.023852 | |
SE | 0.611345 | 0.128798 | 0.439047 | ||
MSE | 0.393664 | 0.016703 | 0.190998 | ||
Nelder–Mead | Estimates | 3.077648 | 4.009598 | 5.024238 | |
SE | 0.609420 | 0.128441 | 0.438062 | ||
MSE | 0.391118 | 0.016610 | 0.190232 | ||
Time | 4d:12h:27m:34s (390454 s) | ||||