Statistical Inference for the power Lindley model based on record values and inter-record times

A new generalization of the Lindley distribution, called the power Lindley distribution was proposed by Ghitany et al., (2013), which offers a more flexible distribution for modeling lifetime data, such as in reliability. They studied classical inferences for the model based on complete data sets. However, we may deal with record breaking data sets in which only values smaller (or larger) than the current extreme value are reported. In this paper, by using record values and inter-record times, we develop inference procedures for the estimation of the parameters and prediction of future record values for the power Lindley distribution. First, the maximum likelihood estimate of the parameters and their asymptotic confidence intervals are obtained. Next, we consider Bayes estimation under the symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions by using the joint bivariate density function. Since the closed forms of the estimates are not available, we encounter some computational difficulties to evaluate the Bayes estimates of the parameters involved in the model. For this reason, we use Tierney and Kadane's method as well as Markov Chain Monte Carlo (MCMC) procedure to compute approximate Bayes estimates. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the power Lindley distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data set is analyzed for illustration purposes.