In this work, we introduce an objective prior based on the kernel density estimation to eliminate the subjectivity of the Bayesian estimation for information other than data. For comparing the kernel prior with the informative gamma prior, the mean squared error and the mean percentage error for the generalized exponential (GE) distribution parameters estimations are studied using both symmetric and asymmetric loss functions via Monte Carlo simulations. The simulation results indicated that the kernel prior outperforms the informative gamma prior. Finally, a numerical example is given to demonstrate the efficiency of the proposed priors.
Published in | Science Journal of Applied Mathematics and Statistics (Volume 12, Issue 2) |
DOI | 10.11648/j.sjams.20241202.12 |
Page(s) | 29-36 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Informative Prior, Kernel Prior, LINEX Loss Function, Squared Error Loss Function
2.1. Informative Kernel Prior
2.2. Informative Gamma Prior
Par. | Kernel Prior | Informative Prior | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
SELF | LLF | SELF | LLF | |||||||||
${\mathit{\theta}}_{\mathit{S}}^{\mathit{*}}$ | MSE | MPE | ${\mathit{\theta}}_{\mathit{L}}^{\mathit{*}}$ | MSE | MPE | ${\mathit{\theta}}_{\mathit{S}}^{\mathit{*}}$ | MSE | MPE | ${\mathit{\theta}}_{\mathit{L}}^{\mathit{*}}$ | MSE | MPE | |
5.1451 | 0.0181 | 0.0255 | 5.0429 | 0.0559 | 0.0448 | 5.2215 | 0.00334 | 0.0109 | 5.0092 | 0.0730 | 0.05119 | |
5.0871 | 0.0371 | 0.0364 | 5.3834 | 0.0108 | 0.0197 | |||||||
0.0501 | 3.17E-04 | 0.5514 | 0.0501 | 3.17E-04 | 0.5514 | 0.0219 | 1.09E-4 | 0.3231 | 0.0218 | 1.09E-4 | 0.3241 | |
0.0501 | 3.17E-04 | 0.5514 | 0.0219 | 1.08E-4 | 0.3221 |
N | $\mathit{\alpha}$ | $\mathit{\beta}$ | Kernel Prior | Informative Prior | ||||||
---|---|---|---|---|---|---|---|---|---|---|
SELF | LLF | SELF | LLF | |||||||
MSE | MPE | MSE | MPE | MSE | MPE | MSE | MPE | |||
20 | 0.5 | 2 | 0.0147 | 0.1889 | 0.0119 | 0.1724 | 0.0399 | 0.2942 | 0.0317 | 0.2616 |
3 | 0.0097 | 0.1596 | 0.0092 | 0.1564 | 0.0240 | 0.2254 | 0.0192 | 0.2035 | ||
1 | 2 | 0.0403 | 0.1624 | 0.0299 | 0.1419 | 0.0994 | 0.2340 | 0.0602 | 0.1850 | |
3 | 0.0332 | 0.1496 | 0.0382 | 0.1629 | 0.0572 | 0.1804 | 0.0399 | 0.1576 | ||
2 | 2 | 0.1047 | 0.1330 | 0.2145 | 0.2160 | 0.1253 | 0.1445 | 0.1581 | 0.1692 | |
3 | 0.2136 | 0.2113 | 0.3493 | 0.2849 | 0.1557 | 0.1662 | 0.2410 | 0.2195 | ||
40 | 0.5 | 2 | 0.0085 | 0.1408 | 0.0076 | 0.1351 | 0.0147 | 0.1795 | 0.0128 | 0.1674 |
3 | 0.0063 | 0.1285 | 0.0062 | 0.1288 | 0.0099 | 0.1480 | 0.0088 | 0.1402 | ||
1 | 2 | 0.0323 | 0.1395 | 0.0262 | 0.1289 | 0.0507 | 0.1672 | 0.0376 | 0.1465 | |
3 | 0.0249 | 0.1294 | 0.0262 | 0.1288 | 0.0348 | 0.1412 | 0.0286 | 0.1314 | ||
2 | 2 | 0.0749 | 0.1115 | 0.1128 | 0.1422 | 0.1136 | 0.1353 | 0.1155 | 0.1417 | |
3 | 0.1222 | 0.1496 | 0.1842 | 0.1946 | 0.1197 | 0.1431 | 0.1526 | 0.1687 | ||
80 | 0.5 | 2 | 0.0042 | 0.1018 | 0.0040 | 0.0998 | 0.0055 | 0.1142 | 0.0051 | 0.1099 |
3 | 0.0036 | 0.0955 | 0.0036 | 0.0961 | 0.0042 | 0.1017 | 0.0039 | 0.0992 | ||
1 | 2 | 0.0191 | 0.1091 | 0.0172 | 0.1049 | 0.0225 | 0.1167 | 0.0190 | 0.1086 | |
3 | 0.0158 | 0.1017 | 0.0163 | 0.1042 | 0.0179 | 0.1059 | 0.0162 | 0.1021 | ||
2 | 2 | 0.0615 | 0.1014 | 0.0718 | 0.1103 | 0.0745 | 0.1106 | 0.0721 | 0.1101 | |
3 | 0.0759 | 0.1140 | 0.0997 | 0.1347 | 0.0746 | 0.1117 | 0.0849 | 0.1208 |
N | $\mathit{\alpha}$ | $\mathit{\beta}$ | Kernel Prior | Informative Prior | ||||||
---|---|---|---|---|---|---|---|---|---|---|
SELF | LLF | SELF | LLF | |||||||
MSE | MPE | MSE | MPE | MSE | MPE | MSE | MPE | |||
20 | 0.5 | 2 | 0.1058 | 0.1307 | 0.2089 | 0.1975 | 0.2378 | 0.1877 | 0.1291 | 0.1465 |
3 | 0.2622 | 0.1385 | 0.4410 | 0.1797 | 0.3327 | 0.1604 | 0.7407 | 0.2669 | ||
1 | 2 | 0.0878 | 0.1202 | 0.1419 | 0.1579 | 0.1784 | 0.1625 | 0.1155 | 0.1386 | |
3 | 0.2205 | 0.1294 | 0.3111 | 0.1506 | 0.2794 | 0.1461 | 0.5410 | 0.2208 | ||
2 | 2 | 0.0910 | 0.1233 | 0.1543 | 0.1734 | 0.0954 | 0.1257 | 0.1137 | 0.1415 | |
3 | 0.2002 | 0.1246 | 0.2443 | 0.1345 | 0.3129 | 0.1597 | 0.5664 | 0.2312 | ||
40 | 0.5 | 2 | 0.0928 | 0.1247 | 0.2213 | 0.1883 | 0.1798 | 0.1657 | 0.1216 | 0.1428 |
3 | 0.1837 | 0.1195 | 0.2467 | 0.1342 | 0.2769 | 0.1453 | 0.4592 | 0.1970 | ||
1 | 2 | 0.0768 | 0.1125 | 0.1739 | 0.1637 | 0.1257 | 0.1402 | 0.0971 | 0.1274 | |
3 | 0.1655 | 0.1142 | 0.1834 | 0.1175 | 0.2192 | 0.1289 | 0.3195 | 0.1600 | ||
2 | 2 | 0.0554 | 0.0963 | 0.1245 | 0.1384 | 0.0718 | 0.1096 | 0.0785 | 0.1161 | |
3 | 0.1555 | 0.1104 | 0.1527 | 0.1082 | 0.1981 | 0.1234 | 0.3045 | 0.1592 | ||
80 | 0.5 | 2 | 0.0799 | 0.1130 | 0.1306 | 0.1411 | 0.1122 | 0.1327 | 0.0875 | 0.1214 |
3 | 0.1525 | 0.1084 | 0.1713 | 0.1129 | 0.1921 | 0.1210 | 0.2504 | 0.1401 | ||
1 | 2 | 0.0597 | 0.0971 | 0.0906 | 0.1171 | 0.0758 | 0.1104 | 0.0644 | 0.1041 | |
3 | 0.1317 | 0.1002 | 0.1317 | 0.0993 | 0.1425 | 0.1040 | 0.1707 | 0.1144 | ||
2 | 2 | 0.0393 | 0.0803 | 0.0664 | 0.1007 | 0.0466 | 0.0887 | 0.0477 | 0.0901 | |
3 | 0.1169 | 0.0932 | 0.1097 | 0.0904 | 0.1132 | 0.0926 | 0.1474 | 0.1068 |
N | $\mathit{\alpha}$ | $\mathit{\beta}$ | Kernel Prior | Informative Prior | ||||||
---|---|---|---|---|---|---|---|---|---|---|
$\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | |||||||
MSE | MPE | MSE | MPE | MSE | MPE | MSE | MPE | |||
20 | 0.5 | 2 | 0.0187 | 0.2100 | 0.1367 | 0.1479 | 0.0512 | 0.3334 | 0.7977 | 0.3477 |
3 | 0.0109 | 0.1667 | 0.3049 | 0.1663 | 0.0306 | 0.2527 | 0.5771 | 0.1951 | ||
1 | 2 | 0.06899 | 0.2076 | 0.1086 | 0.1297 | 0.1797 | 0.3123 | 0.4297 | 0.2515 | |
3 | 0.0362 | 0.1544 | 0.2061 | 0.1289 | 0.0979 | 0.2274 | 0.4169 | 0.1659 | ||
2 | 2 | 0.0581 | 0.0976 | 0.0523 | 0.0914 | 0.3087 | 0.2099 | 0.1339 | 0.1419 | |
3 | 0.1053 | 0.1315 | 0.2291 | 0.1419 | 0.1862 | 0.1693 | 0.2158 | 0.1258 | ||
40 | 0.5 | 2 | 0.0046 | 0.1483 | 0.1367 | 0.1479 | 0.0171 | 0.1931 | 0.3777 | 0.2356 |
3 | 0.0066 | 0.1295 | 0.1771 | 0.1135 | 0.0114 | 0.1573 | 0.4197 | 0.1685 | ||
1 | 2 | 0.0439 | 0.1597 | 0.1086 | 0.1297 | 0.0719 | 0.1969 | 0.2093 | 0.1769 | |
3 | 0.0262 | 0.1299 | 0.1217 | 0.0929 | 0.0465 | 0.1598 | 0.2876 | 0.1412 | ||
2 | 2 | 0.0755 | 0.1111 |