Estimation and prediction for power Lindley distribution under progressively type II right censored samples

Abstract In survival, reliability and medical studies, it is natural to have experience with several situations pertaining to testing, cost or money constraints where the removal of units prior to failure is preplanned. In this context, we consider the inference problem including estimation and prediction for power Lindley distribution under the progressively type-II censored sample data. For the estimation purposes and other reliability characteristics maximum likelihood and Bayes approaches for estimating the model parameters are considered in this paper. Confidence intervals of the parameters and the corresponding average lengths and coverage probabilities are developed based on maximum likelihood and Bayes techniques. The Gibbs and Metropolis samplers are used to predict the life lengths of the removed units in multiple stages of the progressively censored sample. Monte Carlo simulations are performed to compare different methods and one real data set is analyzed for illustrative purposes.

[1]  Norman L. Johnson,et al.  Life Testing and Early Failure , 1966 .

[2]  Miguel A. Sordo,et al.  The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance , 2014 .

[3]  M. E. Ghitany,et al.  Estimation of the Reliability of a Stress-Strength System from Power Lindley Distributions , 2015, Commun. Stat. Simul. Comput..

[4]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Saralees Nadarajah,et al.  A generalized Lindley distribution , 2011 .

[6]  David Lindley,et al.  Fiducial Distributions and Bayes' Theorem , 1958 .

[7]  Debasis Kundu,et al.  Bayesian Inference and Life Testing Plan for the Weibull Distribution in Presence of Progressive Censoring , 2008, Technometrics.

[8]  Kapil Kumar,et al.  Reliability estimation in Lindley distribution with progressively type II right censored sample , 2011, Math. Comput. Simul..

[9]  M. E. Ghitany,et al.  Power Lindley distribution and associated inference , 2013, Comput. Stat. Data Anal..

[10]  Paul I. Nelson,et al.  15 Prediction of order statistics , 1998 .

[11]  N. Balakrishnan,et al.  Progressive Censoring: Theory, Methods, and Applications , 2000 .

[12]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[13]  Ali Dolati,et al.  Generalized Lindley Distribution , 2009 .

[14]  Narayanaswamy Balakrishnan,et al.  Estimation of parameters from progressively censored data using EM algorithm , 2002 .

[15]  N. Balakrishnan,et al.  On some predictors of times to failure of censored items in progressively censored samples , 2006, Comput. Stat. Data Anal..

[16]  Ahmed A. Soliman,et al.  Estimation for the exponentiated Weibull model with adaptive Type-II progressive censored schemes , 2016 .

[17]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[18]  Ming-Hui Chen,et al.  Monte Carlo Estimation of Bayesian Credible and HPD Intervals , 1999 .

[19]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[20]  Sajid Ali,et al.  A study of the effect of the loss function on Bayes Estimate, posterior risk and hazard function for Lindley distribution , 2013 .

[21]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[22]  Ali Karimnezhad,et al.  Bayes, E-Bayes and robust Bayes prediction of a future observation under precautionary prediction loss functions with applications , 2016 .

[23]  Mohammad Z. Raqab,et al.  Inference for the generalized Rayleigh distribution based on progressively censored data , 2011 .

[24]  Luc Devroye,et al.  A simple algorithm for generating random variates with a log-concave density , 1984, Computing.

[25]  M. Z. Raqab,et al.  Statistical inference based on Lindley record data , 2018 .

[26]  Debasis Kundu,et al.  On Progressively Type-II Censored Two-parameter Rayleigh Distribution , 2016, Commun. Stat. Simul. Comput..

[27]  A. Asgharzadeh,et al.  Generalized inverse Lindley distribution with application to Danish fire insurance data , 2017 .

[28]  S. Cabras,et al.  A default Bayesian procedure for the generalized Pareto distribution , 2007 .

[29]  Umesh Singh,et al.  Estimation and prediction for Type-I hybrid censored data from generalized Lindley distribution , 2016 .

[30]  J. Hosking,et al.  Parameter and quantile estimation for the generalized pareto distribution , 1987 .

[31]  Francisco Louzada,et al.  An extended Lindley distribution , 2012 .

[32]  Saralees Nadarajah,et al.  A new weighted Lindley distribution with application , 2016 .

[33]  Hare Krishna,et al.  On progressively first failure censored Lindley distribution , 2016, Comput. Stat..

[34]  Bander Al-Zahrani,et al.  Estimating the parameter of the Lindley distribution under progressive type-II censored data , 2015 .

[35]  Pedro Jodrá,et al.  Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function , 2010, Math. Comput. Simul..

[36]  François Chapeau-Blondeau,et al.  Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2 , 2002, IEEE Trans. Signal Process..

[37]  William J. Padgett,et al.  Parametric and Nonparametric Inference from Record-Breaking Data , 2003 .

[38]  Mohammad Z. Raqab,et al.  Prediction of the remaining testing time for the generalized Pareto progressive censoring samples with applications to extreme hydrology events , 2018 .