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 | 0.0523 | 0.0914 | 0.1990 | 0.1673 | 0.0846 | 0.1162 | |
3 | 0.0809 | 0.1164 | 0.1222 | 0.0961 | 0.1433 | 0.1475 | 0.1619 | 0.1097 | ||
80 | 0.5 | 2 | 0.0045 | 0.1043 | 0.1026 | 0.1249 | 0.0059 | 0.1190 | 0.1745 | 0.1609 |
3 | 0.0036 | 0.0953 | 0.1268 | 0.0959 | 0.0045 | 0.1048 | 0.2561 | 0.1335 | ||
1 | 2 | 0.0222 | 0.1159 | 0.0723 | 0.1045 | 0.0274 | 0.1274 | 0.1019 | 0.1252 | |
3 | 0.01596 | 0.1015 | 0.0945 | 0.0834 | 0.0207 | 0.1125 | 0.1715 | 0.1107 | ||
2 | 2 | 0.0673 | 0.1044 | 0.0387 | 0.0384 | 0.1021 | 0.1256 | 0.0516 | 0.0917 | |
3 | 0.0623 | 0.1017 | 0.0822 | 0.0785 | 0.0835 | 0.1159 | 0.1049 | 0.0887 |
[1] | Gupta RD, Kundu D. Generalized exponential distribution, Australia, and New Zealand Journal of Statistics. 41: 173-188; (1999). |
[2] | Gupta RD, Kundu D. Generalized exponential distribution, Different method of estimations. Journal of statistical computation and simulation. 69: 315-337; (2001a). |
[3] | Gupta RD, Kundu D. Generalized exponential distribution, An alternative to gamma or Weibull distribution. Biometrical journal. 43: 117-130; (2001b). |
[4] | Gupta RD, Kundu D. Generalized exponential distribution statistical inferences. Journal of Statistical theory and applications. 1: 101-118; (2002). |
[5] | Raqab MZ, Ahsanullah M. Estimation of the location and scale parameters of the generalized exponential distributions based on order statistics. Journal of statistical Computation and simulation. 69; 109-123; (2001). |
[6] |
Zheng G. On the fisher information matrix in type-II censored data from the exponentiated with exponential family, Biometrical journal. 44: 353-357; (2002).
https://doi.org/10.1002/1521-4036(200204)44:3<353::AID- BIMJ353>3.0.CO;2-7 |
[7] | Raqab M Z, Ahsanullah M. Inference for generalized exponential distributions based on record statistics. Journal of statistical planning and inference. 104: 339-350; (2002). |
[8] | Raqab MZ, Madi MT. Bayesian inference for the generalized exponential distribution. Journal of Statistical Compution and Simulation. 69(2): 109-124; (2005). |
[9] | Singh R, Singh SK, Singh U, and Singh GP. Bayes Estimator of Generalized-Exponential Parameters under LINEX loss function using Lindley's Approximation. Data Science Journal. 7: 65-75; (2008). |
[10] | Kundu D, Pradhan B. On progressively censored generalized expo nential distribution, Test 18: 497-515; (2009). |
[11] | Sanku Dey. Bayesian Estimation of the Shape Parameter of the Generalised Exponential Distribution under Different Loss Functions. VI (2), (2010). |
[12] | Syed Afzal Hossain. Estimating the Parameters of a Generalized Exponential Distribution. Journal of Statistical Theory and Applications, 17(3): 537-553; (2018). |
[13] | Calabria R, Pulcini G. An engineering approach to Bayes estimation for the Weibull distribution. Microelectron Reliab. 34: 789–802; (1994). |
[14] | Calabria R, Pulcini G. Bayes prediction of number of failures in Weibull samples. Commun. Statist. -Theory Meth. 24(2): 487-499; (1995). |
[15] | Calabria R, Pulcini G. Point estimation under asymmetric loss functions for left-truncated. exponential samples. Commun. Statist. -Theory Meth. 25(3): 585-600; (1996). |
[16] | Abramson I. On Bandwidth variation in kernel estimates: A Square Root Law. Ann. Statist, 10: 1217-1223; (1982). |
[17] | Scott DW. Multivariate density estimation. New York; Wiley- inter-science (1992). |
[18] | Maswadah M. Kernel inference on the inverse Weibull distribution, The Korean Communications in Statistics. 13(3): 503-512; (2006). |
[19] | Maswadah M. Kernel inference on the Weibull distribution. Proc. Of the Third National Statistical Conference. Lahore, Pakistan, May 28-29. 14: 77-86; (2007). |
[20] | Maswadah M. Kernel inference on the type-II Extreme value distribution, Proceedings of the Tenth Islamic Countries Conference on Statistical Sciences (ICCS-X), Lahore, Pakistan. December 10-14, II: 870-880; (2010). |
[21] | Maswadah M. Empirical Bayes inference for the Weibull model. Computational Statistics, 28(6). (2013). |
[22] | Varian HR. A Bayesian approach to real estimate assessment, In: Studies in Bayesian econometrics studies in honor of Leonard J. Savage, North Holand, Amesterdam. 195–208 (1975). |
[23] | Lawless JF. Statistical Models and Methods for Lifetime Data. John Wiley and Sons. New York: 621; (1982). |
APA Style
Maswadah, M., Mohamed, S. (2024). Bayesian Inference on the Generalized Exponential Distribution Based on the Kernel Prior. Science Journal of Applied Mathematics and Statistics, 12(2), 29-36. https://doi.org/10.11648/j.sjams.20241202.12
ACS Style
Maswadah, M.; Mohamed, S. Bayesian Inference on the Generalized Exponential Distribution Based on the Kernel Prior. Sci. J. Appl. Math. Stat. 2024, 12(2), 29-36. doi: 10.11648/j.sjams.20241202.12
AMA Style
Maswadah M, Mohamed S. Bayesian Inference on the Generalized Exponential Distribution Based on the Kernel Prior. Sci J Appl Math Stat. 2024;12(2):29-36. doi: 10.11648/j.sjams.20241202.12
@article{10.11648/j.sjams.20241202.12, author = {Mohamed Maswadah and Seham Mohamed}, title = {Bayesian Inference on the Generalized Exponential Distribution Based on the Kernel Prior }, journal = {Science Journal of Applied Mathematics and Statistics}, volume = {12}, number = {2}, pages = {29-36}, doi = {10.11648/j.sjams.20241202.12}, url = {https://doi.org/10.11648/j.sjams.20241202.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20241202.12}, abstract = {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. }, year = {2024} }
TY - JOUR T1 - Bayesian Inference on the Generalized Exponential Distribution Based on the Kernel Prior AU - Mohamed Maswadah AU - Seham Mohamed Y1 - 2024/05/17 PY - 2024 N1 - https://doi.org/10.11648/j.sjams.20241202.12 DO - 10.11648/j.sjams.20241202.12 T2 - Science Journal of Applied Mathematics and Statistics JF - Science Journal of Applied Mathematics and Statistics JO - Science Journal of Applied Mathematics and Statistics SP - 29 EP - 36 PB - Science Publishing Group SN - 2376-9513 UR - https://doi.org/10.11648/j.sjams.20241202.12 AB - 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. VL - 12 IS - 2 ER -