International Journal of Basic and Applied Sciences, 2 (4) (2013) 346-355
©Science Publishing Corporation
www.sciencepubco.com/index.php/IJBAS
A new bivariate distribution with Gompertz marginals
El-Sherpieny E. A, Ibrahim S.A , Bedar, R.E
Department of Mathematical Statistics, Institute of Statistical Studies & Research, Cairo University, Cairo City, Egypt
*Corresponding author E-mail: ahmedc55@yahoo.com
Abstract
In this paper, we propose a new bivariate distribution with the Gompertz marginals. Some properties of this new
bivariate distribution have been investigated. Several properties of this distribution have been discussed. Parameters
estimation using moments and maximum likelihood methods are obtained. A numerical illustration experiments have
been performed to see the behavior of the MLEs. One data set has been analyzed for illustrative purpose.
Keywords: Bivariate model; Gompertz distribution function; maximum likelihood estimators; moment estimators; fisher information matrix.
1 Introduction
The Gompertz distribution was originally introduced by Gompertz [8].This distribution is widely used to describe
human mortality and establish actuarial tables. It has been used as a growth model and also used to fit the tumor growth.
The Gompertz distribution is related by a simple transformation to certain distribution in the family of distributions
obtained by Pearson. Applications and more recent survey of the Gompertz distribution can be introduced by Ahuja and
Nash [2].
In many practical problems, multivariate lifetime data arise frequently, and in these situations it is important to consider
different multivariate models that could be used to model such multivariate lifetime data. For an encyclopedia treatment
on various multivariate models and their properties and applications, one may refer to the book by Kotz et al. [9].
In fact, shock models are used in reliability to describe different applications. Shocks can refer for example to damage
caused to biological organs by illness or environmental causes of damage acting on a technical system, El-Gohary and
Sarhan [6], and A-hameed and Proschan [1]. Also Al-Ruzaiza and El-Gohary have obtained a new class of bivariate
distribution with Pareto of Marshall-Olkin type [4].
The objective of this paper is to introduce a new bivariate Gompertz distribution of Marsall-Olkin type. It is considered
as a distribution of the life times of two dependent components each has a Gompertz distribution. Also discuss about the
computation of the maximum likelihood estimators and moment generating function.
The paper is organized as follows. Section 2 presents the shock model yielding the new bivariate Gompertz distribution.
The joint survival and probability density function of bivariate Gompertz distribution is obtained. Section 3 presents the
joint moment generating function of this bivariate distribution and its marginal moment generating functions. Section 4
discusses the maximum likelihood estimation of proposed new bivariate Gompertz distributions. Section 5 presents the
simulation and one data analysis results. Finally we conclude the paper in section 6.
2 The new bivariate Gompertz distribution
We define a new bivariate Gompertz distribution (), shortly denoted by distribution. We start with the joint
survival function of the distribution and then derive the corresponding joint probability density function.
2.1 The Joint Survival Function
It is assumed that the univariate Gompertz distribution with the shape parameter α>0 and the scale parameter λ>0 has
the following probability density function, cumulative distribution function and survival function for x>0;
(1)