On a Mixture of the Lindley and Modified Lindley Distributions: Properties and Estimation

Abstract

In this article, we investigate a novel three-parameter lifetime distribution constructed from a mixture of the original Lindley and modified Lindley distributions. The concept behind this construction is to combine the contrasting properties of these two well-known distributions to provide a new statistical modeling option for lifetime data analysis. In particular, it provides a natural alternative to the three-parameter, two-component mixture of the Lindley distribution, which has attracted attention in the recent statistical literature. We investigate its main properties from both a theoretical and practical point of view. The shapes of the corresponding probability density and hazard rate functions and the formulas for the moments, moment generating functions and characteristic functions are discussed. The distribution is then subjected to statistical analysis, considering it as a semi-parametric model. The maximum likelihood approach is used to estimate the parameters. In a simulation analysis, the numerical behavior of the bias and the mean square error of the obtained estimates are studied. The new  model is tested on three data sets and the results show that it has a better fit behavior than its main competitor, the three-parameter two-component mixture of the Lindley model.

Keywords: Lindley distributions; maximum likelihood estimates; mixture distribution; modified Lindley distribution; moments.

2010 Mathematics Subject Classification. 26A25; 26A35