On the Bayesian estimation of the weighted Lindley distribution. (English) Zbl 1457.62303
Summary: The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose.
MSC:
| 62N05 | Reliability and life testing |
| 62E15 | Exact distribution theory in statistics |
| 62F15 | Bayesian inference |
| 62-08 | Computational methods for problems pertaining to statistics |
Keywords:
weighted Lindley distribution; loss functions; posterior risk; failure rate function; mean residual life function; Lindley approximation; importance samplingReferences:
| [1] | Patil GP. Weighted distributions. In: El-ShaarawiAH, PiegorschWW, editors. Encyclopedia of environmetrics. Chichester: Wiley; 2002;4:2369-2377. [Google Scholar] |
| [2] | Lindley DV. Fiducial distributions and Bayes theorem. J R Stat Soc, Ser. B (Methodological). 1958;9(1):102-107. [Google Scholar] · Zbl 0085.35503 |
| [3] | Ghitany , ME, Atieh B, Nadarajah S. Lindley distribution and its application. Math Comput Simul. 2008;78:493-506. [Crossref], [Google Scholar] · Zbl 1140.62012 |
| [4] | Sankaran M. The discrete Poisson-Lindley distribution. Biometrics. 1970;26:145-149. [Google Scholar] |
| [5] | Ghitany ME, Al-Mutairi DK. Estimation methods for the discrete Poisson-Lindley distribution. J Stat Comput Simul. 2009;79(1):1-9. [Taylor & Francis Online], [Google Scholar] · Zbl 1161.62010 |
| [6] | Ghitany ME, Al-Mutairi DK, Nadarajah S. Zero-truncate Poisson-Lindley distribution and its application. Math Comput Simul. 2008;79:279-287. [Crossref], [Google Scholar] · Zbl 1153.62308 |
| [7] | Ghitany ME, Al-Mutairi DK. Size-biased Poisson-Lindley distribution and its application. METRON - Int J Stat 2008;LXVI (3), 299-311. [Google Scholar] · Zbl 1416.62107 |
| [8] | Zamani H, Ismail N. Negative Binomial-Lindley distribution and its application. J Math Stat. 2010;6(1):4-9. [Crossref], [Google Scholar] · Zbl 1188.91092 |
| [9] | Ghitany ME, Alqallaf F, Al-Mutairi DK, Husain HA. A two-parameter weighted Lindley distribution and its applications to survival data. Math Comput Simul. 2011;81(6):1190-1201. [Crossref], [Google Scholar] · Zbl 1208.62021 |
| [10] | Shanker R, Sharma S, Shanker, R.A two-parameter Lindley distribution for modeling waiting and survival times data. Appl Math. 2013;4 (2):363-368. doi: 10.4236/am.2013.42056 [Crossref], [Google Scholar] |
| [11] | Jeffreys H. Theory of probability. 3rd ed. Oxford: Oxford University Press; 1964. [Google Scholar] · Zbl 0023.14501 |
| [12] | Bansal AK. Bayesian parametric inference. New Delhi: Narosa Publishing House Pvt. Ltd.; 2007. [Google Scholar] |
| [13] | Ali S, Aslam M, Kazmi SMA. A study of the effect of the loss function on Bayes estimate, posterior risk and hazard function for Lindley distribution, Appl Math Model. 2013;37(8):6068-6078. [Crossref], [Google Scholar] · Zbl 1274.62194 |
| [14] | Lindley DV. Approximate Bayes methods. Trabajos de Estadistica Y de Investigacion Operativa. 1980;31(1): 223-245. [Google Scholar] |
| [15] | Howlader HA, Hossain A. Bayesian survival estimation of Pareto distribution of the second kind based on failure censored data. Comput Stat Data Anal. 2002;38:301-314. [Crossref], [Google Scholar] · Zbl 1079.62505 |
| [16] | Singh R, Singh SK, Singh U, Singh GP. Bayes estimator of generalized exponential parameters under Linex loss function using Lindley’s approximation. Data Sci J. 2008;7:65-75. [Crossref], [Google Scholar] · Zbl 1274.62210 |
| [17] | Preda V, Panaitescu E, Constantinescu A. Bayes estimators of modified-Weibull distribution parameters using Lindley’s approximation. WSEAS Trans Math. 2010;9(7):539-549. [Google Scholar] · Zbl 1231.62007 |
| [18] | Kundu D, Pradhan B. Bayesian inference and life testing plans for generalized exponential distribution. Sci China Ser A: Math. 2009;52(6):1373-1388. [Crossref], [Google Scholar] · Zbl 1176.62024 |
| [19] | Devroye L. A simple algorithm for generating random variates with log-concave density. Computing. 1984;33: 247-257. [Crossref], [Google Scholar] · Zbl 0561.65004 |
| [20] | Guérin F, Dumon B, Usureau E. Reliability estimation by Bayesian method: definition of prior distribution using dependability study. Reliab Eng Syst Saf. 2003;82:299-306. [Crossref], [Google Scholar] |
| [21] | Shaked M, Shanthikumar JG. Stochastic orders and their applications. Boston, MA: Academic Press; 1994. [Google Scholar] · Zbl 0806.62009 |
| [22] | Giorgiand GM, Nadrajah S. Bonferroni and Gini indices for various parametric families of distributions. METRON - Int J Stat. 2010;LXVIII(1):23-46. [Google Scholar] · Zbl 1301.91042 |
| [23] | Gupta RC, Gupta RD. Proportional reversed hazard model and its applications. J Stat Plan Inference. 2007;137(1):3525-3536. [Crossref], [Google Scholar] · Zbl 1119.62098 |
| [24] | Barlow RE, Proschan F. Statistical theory of reliability and life-testing. New York, NY: Holt, Rinehart, and Winston; 1975. [Google Scholar] · Zbl 0379.62080 |
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