Elsevier

Statistical Methodology

Volume 9, Issue 6, November 2012, Pages 589-603
Statistical Methodology

On the discrete analogues of continuous distributions

https://doi.org/10.1016/j.stamet.2012.03.003Get rights and content

Abstract

In this paper, a new method is proposed for generating discrete distributions. A special class of the distributions, namely, the T-geometric family contains the discrete analogues of continuous distributions. Some general properties of the T-geometric family of distributions are obtained. A member of the T-geometric family, namely, the exponentiated-exponential–geometric distribution is defined and studied. Various properties of the exponentiated-exponential–geometric distribution such as the unimodality, the moments and the probability generating function are discussed. The method of maximum likelihood estimation is proposed for estimating the model parameters. Three real data sets are used to illustrate the applications of the exponentiated-exponential–geometric distribution.

Introduction

Discrete distributions are very important for modeling real life scenarios. Many research papers have been published on the study and applications of discrete distributions. A large number of discrete distributions can be found in [10], [3], [5]. Various techniques for generating families of discrete distributions have been developed.

Katz [11] developed a discrete analogue of the Pearson continuous system by using the relationship px+1px=a+bx1+x,x=0,1,2,. A more general extension of the Katz family is the Kemp families of distributions. The family of generalized hypergeometric probability distributions by Kemp [12] generated a wide variety of existing discrete distributions. For detailed discussion, one may refer to Johnson et al. [10, Chapter 2].

The Lagrangian family of discrete distributions, generated by using the Lagrangian expansion, is another important technique for generating discrete distributions, which was studied by Consul, Shenton and their collaborators beginning in the early 1970s. For detailed discussion, one may refer to the books by Consul [4] and Consul and Famoye [5]. More recently, Li et al. [20] relaxed some conditions on Lagrangian expansion and used it to generate the family of generalized Lagrangian probability distributions.

Some discrete analogues of continuous distributions were developed by using the form P(X=k)=f(k)/u=f(u),k=0,±1,±2,, where f is the probability density function of a continuous random variable. Examples include the discrete normal distribution studied by Kemp [13] and the discrete Laplace distribution studied by Inusah and Kozubowski [9].

Roknabadi et al. [22] defined the telescopic family of distributions as the one with probability function P(X=x)=qkθ(x)qkθ(x+1),x=0,1,2,, where 0<q<1 and kθ(x) is strictly increasing function of x with kθ(0)=0 and kθ(x) as x. They showed that if a continuous random variable belongs to the extended exponential family (e.g. exponential, Rayleigh and Weibull) with cumulative distribution function G(t)=1exp[αkθ(t)], then the discrete versions of these continuous distributions are members of the telescopic family of distributions.

In his study of discrete reliability measures, Roy [23] pointed out that geometric distribution is a discrete analogue of exponential distribution. Subsequently, Roy [24], [25] proposed a method from reliability perspective to discretize continuous distributions and studied discrete analogues of Rayleigh and normal distributions. Denote the survival function of a continuous random variable X by S(x)=P(Xx). If times are grouped into unit intervals, the discrete observed variable dX=[X], the largest integer less than or equal to X, has the probability function P(dX=x)=p(x)=P(xX<x+1)=S(x)S(x+1),x=0,1,2,. Krishna and Pundir [18] applied this method to study the discrete Burr and discrete Pareto distributions. The result in (1.1) belongs to the telescopic family of Roknabadi et al. [22].

In this paper, we propose another technique to generate new families of discrete distributions. A new discrete distribution generated using this technique is studied in detail. In Section 2, we define and study some properties of the family of discrete analogue of the distribution for any non-negative continuous random variable, namely, the T-geometric distribution. In Section 3, a member of T-geometric family, the exponentiated-exponential–geometric distribution (EEGD) is defined and studied. In Section 4, the method of maximum likelihood estimation (MLE) is proposed to estimate the EEGD parameters. In Section 5, the likelihood ratio test, the Wald test and the score test are proposed to compare the geometric distribution with the EEGD. A simulation study is conducted to evaluate the performance of the three tests. Applications of the EEGD to real data sets are provided in Section 6.

Section snippets

Discrete analogues of distributions of non-negative continuous random variables

Let F(x) be the cumulative distribution function (CDF) of any random variable X and let r(t) be the probability density function of a continuous random variable T defined on [0,). The CDF of the T-X family of distributions defined by Alzaatreh et al. [1] is given as G(x)=0log(1F(x))r(t)dt=R{log(1F(x))}, where R(t) is the CDF of the random variable T. Alzaatreh et al. [1] in their paper call the family of distributions defined in (2.1) the ‘Transformed-Transformer’ family (or T-X family).

Definition and some properties of the EEGD

Gupta and Kundu [7] defined and studied the exponentiated-exponential distribution. If a random variable T follows the exponentiated-exponential distribution, then its CDF is given by R(t)=(1eλt)α,t>0,α>0,λ>0. If X is a random variable that follows the geometric distribution in (2.4), then the T-X family in (2.2) leads to the exponentiated-exponential–geometric distribution with the pmf g(x)=(1pλ(x+1))α(1pλx)α,x=0,1,2,. On replacing pλ by θ, (3.1) can be written as g(x)=(1θx+1)α(1θx)α,x

Parameter estimation for EEGD

The maximum likelihood method is applied to estimate the EEGD parameters. Let a random sample of size n be taken from the EEGD and let the observed frequencies be denoted by nx,x=0,1,2,,k, where Σx=0knx=n. The log-likelihood function of the EEGD in (3.2) can be written as logL=n0αlog(1θ)+x=1knxlog{(1θx+1)α(1θx)α}. The derivatives of (4.1) with respect to α and θ are, respectively, given by logLα=n0log(1θ)+x=1k[(1θx+1)αlog(1θx+1)(1θx)αlog(1θx)]nx(1θx+1)α(1θx)α,logLθ=n0α1θ+x

Tests to compare EEGD with geometric distribution

The EEGD reduces to the geometric distribution when α=1. Thus, to compare the EEGD with the geometric distribution, we test the null hypothesis H0:α=1againstH1:α1. To test the null hypothesis in (5.1), one can use the likelihood ratio test, the Wald test, or the score test. The likelihood ratio statistic for testing the null hypothesis in (5.1) is based on λ=L0(θ̃)/L1(αˆ,θˆ), where L0 and L1 are the likelihood functions for the geometric distribution and the EEGD respectively. The likelihood

Applications of EEGD

In this section, the EEGD is applied to three data sets. The first data set in Table 4 is from [4, p. 120] and it represents the observed frequencies of the number of outbreaks of strike in a coal-mining industry in the UK during 1948–1959. The second data set from [8], in Table 5, represents the observed frequencies of the number of absences among shift-workers in a steel industry. The third data set, in Table 6, from [17, p. 135] represents the observed frequencies of the number of claims on

Summary

A method to generate new families of discrete distributions is introduced. Some examples of discrete distributions were provided. Attention is devoted to a special class of the T-X families of discrete distributions when X is the geometric distribution. Some properties of the T-geometric family of distributions are obtained. A member of the T-geometric family of discrete distributions, the exponentiated-exponential–geometric distribution, is defined. Various properties of the EEGD are studied,

Acknowledgments

The authors are grateful for the comments and suggestions by the referees and the Editor-in-Chief. Their comments and suggestions have greatly improved the paper.

References (27)

  • P.C. Consul et al.

    Lagrangian Probability Distributions

    (2006)
  • D.R. Cox et al.

    Theoretical Statistics

    (1974)
  • R.D. Gupta et al.

    Exponentiated-exponential family: an alternative to gamma and Weibull distributions

    Biometrical Journal

    (2001)
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