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

Subject Areas : International Journal of Mathematical Modelling & Computations

Christophe Chesneau ¹ , Haitham Yousof ² , G. Hamedani ³ , Mohamed Ibrahim ⁴

1 - Department of Mathematics, LMNO, University of Caen, France
2 - Department of Statistics,Mathematics and Insurance, Benha University, Benha, Egypt
3 - Department of Mathematics, Statistics and Computer Science, Marquette University, USA
4 - Department of Applied Statistics and Insurance, Faculty of Commerce, Damietta University, Damietta, Egypt

Received: 2020-10-03 Accepted : 2020-11-16 Published : 2021-03-01

Keywords: COPULA, Bayesian Estimation, Different Methods of Estimations, Markov chain Monte Carlo Simulations, Cramer-Von-Mises, Lindley Distribution, characterizations,

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.