Improved maximum likelihood estimators for the parameters of the two parameter lindley distribution

Abstract

Two-parameter Lindley (TPL) distribution is becoming increasingly popular for modeling lifetime and survival times data, while maximum likelihood estimators (MLEs) are biased for small and moderate sample sizes. This problem has been a motivation to obtain nearly unbiased estimators for the parameters of the model. For this purpose, for the first time, two different techniques, the Cox-Snell methodology, and Efron’s bootstrap method, have been used to improve modified nearly unbiased estimators for MLEs of the unknown parameters of the TPL distribution. A Monte Carlo simulation study has been performed to compare the performance of these proposed techniques with different sample sizes and parameter values. In the simulation study, bias and mean square error (MSE) criteria were taken into consideration as evaluation criteria. In addition, a real example is given to demonstrate the applicability of the techniques. The numerical results show that the bias-corrected estimators outperform the other estimators in terms of biases and mean square errors.

Keywords

• Bootstrap Bias-Correction
• CoxSnell Bias-Correction
• Maximum Likelihood Estimators
• MonteCarlo Simulation
• TwoParameter Lindley Distribution

References

1. REFERENCES
2. [1] Lindley DV. Fiducial distributions and Bayes’ theorem. J Royal Stat Soc Series B (Methodological) 1958;20:102–107. [CrossRef]
3. [2] Ghitany ME, Atieh B, Nadarajah S. Lindley distribution and its application. Mathematics and computers in simulation 2008;78:493–506. [CrossRef]
4. [3] Mazucheli J, Achcar JA. The Lindley distribution applied to competing risks lifetime data. Computer methods and programs in biomedicine 2011;104:188–192. [CrossRef]
5. [4] Krishna H, Kumar K. Reliability estimation in Lindley distribution with progressively type II right censored sample. Math Comput Simul 2011;82:281–294. [CrossRef]
6. [5] Al-Mutairi DK, Ghitany ME, Kundu D. Inferences on stress-strength reliability from Lindley distributions. Commun Stat Theory Methods 2013;42:1443–1463. [CrossRef]
7. [6] Gupta PK, Singh B. Parameter estimation of Lindley distribution with hybrid censored data. Int J Syst Assur Eng Manag 2013;4:378–385. [CrossRef]
8. [7] Al-Zahrani B, Ali MA. Recurrence relations for moments of multiply type-II censored order statistics from Lindley distribution with applications to inference. Stat Optim Inform Comput 2014;2:147–160. [CrossRef]

9. [8] Jia J, Song H. Parameter estimation of lindley distribution under generalized first-failure progressive hybrid censoring schemes. IAENG Int J Appl Math 2022;52:799.
10. [9] Shanker R, Kamlesh KK, Fesshaye H. A two parameter Lindley distribution: Its properties and applications. Biostat Biom Open Access J 2017;1:85–90. [CrossRef]
11. [10] Cox DR, Snell EJ. A general definition of residuals. Journal of the Royal Statistical Society: Series B (Methodological) 1968;30:248–265. [CrossRef]
12. [11] Efron B. The jackknife, the bootstrap and other resampling plans. CBMS-NSF Regional Conference Series in Applied Mathematics, Monograph 38, SIAM, Philadelphia.
13. [12] Cordeiro GM, Da Rocha EC, Da Rocha JGC, Cribari-Neto F. Bias-corrected maximum likelihood estimation for the beta distribution. J Stat Comput Simul 1997;58:21–35. [CrossRef]
14. [13] Cribari-Neto F, Vasconcellos KL. Nearly unbiased maximum likelihood estimation for the beta distribution. J Stat Comput Simul 2002;72:107–118. [CrossRef]
15. [14] Saha K, Paul S. Bias‐corrected maximum likelihood estimator of the negative binomial dispersion parameter. Biometrics 2005;61:179–185. [CrossRef]
16. [15] Lemonte AJ, Cribari-Neto F, Vasconcellos KL. Improved statistical inference for the two-parameter Birnbaum-Saunders distribution. Comput Stat Data Anal 2007;51:4656–4681. [CrossRef]
17. [16] Lemonte AJ. Improved point estimation for the Kumaraswamy distribution. J Stat Comput Simul 2011;81:1971–1982. [CrossRef]
18. [17] Giles DE. Bias reduction for the maximum likelihood estimators of the parameters in the half-logistic distribution. Commun Stat Theory Methods 2012;41:212–222. [CrossRef]
19. [18] Giles DE, Feng H, Godwin RT. On the bias of the maximum likelihood estimator for the two-parameter Lomax distribution. Commun Stat Theory Methods 2013;42:1934–1950. [CrossRef]
20. [19] Ling X, Giles DE. Bias reduction for the maximum likelihood estimator of the parameters of the generalized Rayleigh family of distributions. Commun Stat Theory Methods 2014;43:1778–1792. [CrossRef]
21. [20] Giles DE, Feng H, Godwin RT. Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution. Commun Stat Theory Methods 2016;45:2465–2483 [CrossRef]
22. [21] Schwartz J, Giles DE. Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution. Commun Stat Theory Methods 2016;45:465–478. [CrossRef]
23. [22] Wang M, Wang W. Bias-corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution. Commun Stat Simul Comput 2017;46:530–545. [CrossRef]
24. [23] Reath J, Dong J, Wang M. Improved parameter estimation of the log-logistic distribution with applications. Comput Stat 2018;33:339–356. [CrossRef]
25. [24] Mazucheli J, Dey S. Bias-corrected maximum likelihood estimation of the parameters of the generalized half-normal distribution. J Stat Comput Simul 2018;88:1027–1038. [CrossRef]
26. [25] Mazucheli J, Menezes AFB, Dey S. Improved maximum- likelihood estimators for the parameters of the unit-gamma distribution. Commun Stat Theory Methods 2018;47:3767–3778. [CrossRef]
27. [26] Mazucheli J, Menezes AFB, Dey S. Bias-corrected maximum likelihood estimators of the parameters of the inverse Weibull distribution. Commun Stat Simul Comput 2019;48:2046–2055. [CrossRef]
28. [27] Menezes AFB, Mazucheli J. Improved maximum likelihood estimators for the parameters of the Johnson SB distribution. Commun Stat Simul Comput 2020;49:1511–1526. [CrossRef]
29. [28] Menezes A, Mazucheli J, Alqallaf F, Ghitany ME. Bias-corrected maximum likelihood estimators of the parameters of the unit-weibull distribution. Austrian Journal of Statistics 2021;50:41–53. [CrossRef]
30. [29] Tsai TR, Xin H, Fan YY, Lio Y. Bias-Corrected Maximum Likelihood Estimation and Bayesian Inference for the Process Performance Index Using Inverse Gaussian Distribution. Stats 2022;5:1079– 1096. [CrossRef]
31. [30] Dey S, Wang L. Methods of estimation and bias corrected maximum likelihood estimators of unit burr III distribution. Am J Math Manag Sci 2022;41:316–333. [CrossRef]
32. [31] Nelder JA, Mead R. A simplex method for function minimization. Comput J 1965;7:308–313. [CrossRef]
33. [32] Cordeiro GM, Klein R. Bias correction in ARMA models. Stat Probab Lett 1994;19:169–176. [CrossRef]
34. [33] Efron B, Tibshirani RJ. An introduction to the bootstrap. Boca Raton London New York Washington, D.C.: Chapman & Hall/CRC; 1993.
35. [34] Davison AC, Hinkley DV. Bootstrap methods and their application (No. 1). Cambridge: Cambridge University Press; 1997. [CrossRef]
36. [35] Lawless JF. Statistical models and methods for lifetime data. Hoboken, New Jersey: John Wiley & Sons; 2011.