A New Bivariate Distribution Obtained by Compounding the Bivariate Normal and Geometric Distributions

Eisa Mahmoudi,Hamed Mahmoodian

doi:10.2991/jsta.2017.16.2.5

Abstract

Recently, Mahmoudi and Mahmoodian [7] introduced a new class of distributions which contains univariate normal–geometric distribution as a special case. This class of distributions are very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the normal–geometric distribution marginals. Different properties of this new bivariate distribution have been studied. This distribution has five unknown parameters. The EM algorithm is used to determine the maximum likelihood estimates of the parameters. We analyze one series of real data set for illustrative purposes.

Highlights

In recent years, different techniques for extending the family of normal distributions have been proposed
We propose to use EM algorithm to compute the maximum likelihood estimators (MLEs) of the unknown parameters
Some of the points are quite clear from the simulation results: (i) Convergence has been achieved in all cases and this emphasizes the numerical stability of the EM algorithm. (ii) The differences between the average estimates and the true values are almost small. (iii) These results suggest that the EM estimates have performed consistently. (iv) As the sample size increases, the standard errors of the MLEs decrease

Summary

Introduction

Different techniques for extending the family of normal distributions have been proposed. The normal–geometric distribution is defined as follows: The univariate random variable X is said to have a normal–geometric (NG). We introduce bivariate normal–geometric distribution with the normal–geometric distribution marginals. Different properties of this new distribution have been investigated. We shall use the following notation throughout this paper: φ (·) and Φ(·) for the standard normal probability density and cumulative distribution function, respectively, φn(· ; μ , Σ) for the pdf of Nn(μ , Σ) (n -variate normal distribution with mean vector μ and covariance matrix Σ, Φn(· ; μ , Σ) for the cdf of Nn(μ , Σ) (in both singular and non-singular cases), Φn(· ; Σ) for the case when μ = 0.

Properties

Inference

Simulation

Application

Conclusion