The power-linear hazard rate distribution and estimation of its parameters under progressively type-II censoring

Year 2019, Volume: 48 Issue: 3, 818 - 844, 15.06.2019

Bahman Tarvirdizade , Nader Nematollahi

Abstract

In this paper, we introduce a class of distributions which generalizes the power hazard rate distribution and is obtained by combining the linear and power hazard rate functions. This class of distributions, which is called the power-linear hazard rate distribution, is simple and flexible and contains some important lifetime distributions. The maximum likelihood estimators of the parameters using the Newton-Raphson (NR) and the expectation-maximization (EM) algorithms and the Bayes estimators of the parameters under squared error loss (SEL), linear-exponential (LINEX) and Stein loss functions are obtained based on progressively type-II censored sample. Also, we obtain the asymptotic confidence interval and some bootstrap confidence intervals and construct the HPD credible interval for the parameters. A real data set is analyzed and observed that the present hazard rate distribution can provide a better fit than other three-parameter distributions. Finally, a Monte Carlo simulation study is conducted to investigate and compare the performance of different types of estimators presented in this paper.

Keywords

Bayes estimation, Bootstrap confidence intervals, EM algorithm, MCMC methods, Power-linear hazard rate distribution, Progressive type-II censoring

References

• Akdam, N., Kinaci, I. and Saracoglu, B. Statistical inference of stress-strength reliability for the exponential power distribution based on progressive type-II censored samples, Hacettepe Journal of Mathematics and Statistics 46, 239–253, 2017.
• Bain, L. Analysis for the linear failure-rate life testing distribution, Technometrics 15, 551– 559, 1974.
• Balakrishnan, N. and Aggarwala, R. Progressive Censoring, Theory, Methods and Applica- tions, Birkhauser, Boston, 2000.
• Balakrishnan, N. and Sandhu, R.A. A simple simulation algorithm for generating progres- sively type-II generated samples, American Statistician 49, 229–230, 1995.
• Chen, M.H. and Shao, Q.M. Monte Carlo estimation of Bayesian credible and HPD inter- vals, Journal of Computational and Graphical Statistics 8, 69–92, 1999.
• Cohen, A.C. Progressively censored samples in life testing, Technometrics 5, 327–339, 1963.
• Cordeiro G.M., Ortega, E.M. and Popovic, B.V. The gamma-linear failure rate distribution: theory and applications, Journal of Statistical Computation and Simulation 84, 2408–2426, 2014.
• Dempster, A.P., Laird, N.M. and Rubin, D.B. Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. Ser. B 39, 1–38, 1977.
• Efron, B. and Tibshirani, R.J. An Introduction to the Bootstrap, Chapman and Hall, New York, 1993.
• Gelfand, A.E. and Smith, A.F.M. Sampling-based approaches to calculating marginal den- sities, Journal of the American Statistical Association 85, 398–409, 1990.
• Ghitany, M.E. and Kotz, S. Reliability properties of extended linear failure rate distributions, Probability in the Engineering and Informational Sciences 21, 441–450, 2007.
• Jiang, H., Xie, M. and Tang, L.C. On MLEs of the parameters of a modified Weibull distribution for progressively type-II censored samples, Journal of Applied Statistics 37, 617–627, 2010.
• Kinaci, I. Estimation of $P(Y<X)$ for distributions having power hazard function, Pakistan Journal of Statistics 30, 57–70, 2014.
• Kus, C. and Kaya, M.F. Estimation of parameters of the loglogistic distribution based on progressive censoring using the EM algorithm, Hacettepe Journal of Mathematics and Sta- tistics 35, 203–211, 2006.
• Lin, C.T., Wu. S.J.S. and Balakrishnan, N. Parameter estimation for the linear hazard rate distribution based on records and inter-record times, Communications in Statistics–Theory and Methods 32, 729–748, 2003.
• Lin, C.T., Wu. S.J.S. and Balakrishnan, N. Monte Carlo methods for Bayesian inference on the linear hazard rate distribution, Communications in Statistics–Simulation and Com- putation 35, 575–590, 2006.
• Linhart, H. and Zucchini, W. Model Selection, Wiley, New York, 1986.
• Makeham, W.M. On the law of mortality and the construction of annuity tables, J. Inst. Actuar. Assur. Mag. 8, 301–310, 1860.
• Miller, R.G. Survival Analysis, John Wiley, New York, 1981.
• Mugdadi, A.R. The least squares type estimation of the parameters in the power hazard function, Applied Mathematics and Computation 169, 737–748, 2005.
• Mugdadi, A.R. and Min, A. Bayes estimation of the power hazard function, Journal of Interdisciplinary Mathematics 12, 675–689, 2009.
• Ng, T., Chan, C.S. and Balakrishnan, N. Estimation of parameters from progressively cen- sored data using EM algorithm, Computational Statistics and Data Analysis 39, 371–386, 2002.
• Pandey, A., Singh, A. and Zimmer, W.J. Bayes estimation of the linear hazard rate model, IEEE Transactions on Reliability 42, 636–640, 1993.
• Parsian, A. and Nematollahi, N. Estimation of scale parameter under entropy loss function, Journal of Statistical Planning and Inference 52, 377–391, 1996.
• Rastogi, M.K., Tripathi, Y.M. and Wu, S.J. Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring, Journal of Applied Statistics 39, 2389–2411, 2012.
• Rastogi, M.K. and Tripathi, Y.M. Parameter and reliability estimation for an exponen- tiated half-logistic distribution under progressive type-II censoring, Journal of Statistical Computation and Simulation 84, 1711–1727, 2014.
• Rinne, H. The hazard rate: theory and inference, Justus-Liebig University Press, 2014.
• Sarhan, A.M. and Kundu, D. Generalized linear failure rate distribution, Communications in statistics: Theory and Methods 38, 642–666, 2009.
• Sarhan, A.M. and Zaindin, M. Modified Weibull distribution, Applied Sciences 11, 123–136, 2009.
• Sen, A. and Bhattacharyya, G. Inference procedures for the linear failure rate model, Journal of Statistical Planning and Inference 46, 59–76, 1995.
• Sen, A., Kannan, N. and Kundu, D. Bayesian planning and inference of a progressively censored sample from linear hazard rate distribution, Computational Statistics and Data Analysis 63, 108-121, 2013.
• Tian, Y., Tian, M. and ZHU, Q. A new generalized linear exponential distribution and its applications, Acta Mathematicae Applicatae Sinica, English Series 30, 1049–1062, 2014.
• Tierney, L. Markov chains for exploring posterior distributions, The Annals of Statistics 22, 1701–1728, 1994.
• Varian, H. A Bayesian approach to real estate assessment, Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savege, Amsterdam: North Holland 195–208, 1975.