The Lindley-Lindley Distribution: Characterizations, Copula, Properties, Bayesian and Non-Bayesian Estimation

Abstract

‎A new continuous distribution called Lindley-Lindley distribution is defined‎ ‎and studied‎. ‎Relevant mathematical properties are derived‎. ‎We‎ ‎present three characterizations of the new distribution based on the truncated moments of certain functions of the random variable;‎ ‎the hazard function and in terms of the conditional expectation of a‎‎function of the random variable‎. ‎Some new bivariate type distributions using‎ ‎Farlie Gumbel Morgenstern copula‎, ‎modified Farlie Gumbel Morgenstern‎ ‎copula and Clayton copula are introduced‎. ‎The main‎ ‎justification of this paper is to show how different frequentist estimators‎ ‎of the new model perform for different sample sizes and different parameter‎‎values and to provide a guideline for choosing the best estimation method‎ ‎for the parameters of the proposed model‎. ‎The unknown parameters of the new‎ ‎distribution are estimated using the maximum likelihood‎, ‎ordinary‎ ‎least squares‎, ‎Cramer-Von-Mises‎, ‎weighted least squares and Bayesian methods‎. ‎The obtained estimators are compared using‎ ‎Markov Chain Monte Carlo simulations and observed that Bayesian estimators‎ ‎are generally more efficient than the other estimators.