A discrete distribution including the Poisson

Abstract

This article presents a new generalization of the Poisson distribution, with the parameters α > 0 and θ > 0, using the Marshall and Olkin (1997 Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84(3):641–652.[Crossref], [Web of Science ®] , [Google Scholar]) scheme and adding a parameter to the classical Poisson distribution. The particular case of α = 1 gives the Poisson distribution. The new distribution is unimodal and has a failure rate that monotonically increases or decreases depending on the value of the parameter α. After reviewing some of the properties of this distribution, we investigated the question of parameter estimation. Expected frequencies were calculated for two data sets, one with an index of dispersion larger than one and the other with an index of dispersion smaller than one. In both cases the distribution provided a very satisfactory fit.     

An extended Poisson distribution

ABSTRACT

The Poisson distribution is extended over the set of all integers. The motivation comes from the many reflected versions of the gamma distribution, the continuous analog of the Poisson distribution, defined over the entire real line. Various mathematical properties of the extended Poisson distribution are derived. Estimation procedures by the methods of moments and maximum likelihood are also derived with their performance assessed by simulation. Finally, a real data application is illustrated.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

A new discrete distribution: properties and applications in medical care

This paper proposes a simple and flexible count data regression model which is able to incorporate overdispersion (the variance is greater than the mean) and which can be considered a competitor to the Poisson model. As is well known, this classical model imposes the restriction that the conditional mean of each count variable must equal the conditional variance. Nevertheless, for the common case of well-dispersed counts the Poisson regression may not be appropriate, while the count regression model proposed here is potentially useful. We consider an application to model counts of medical care utilization by the elderly in the USA using a well-known data set from the National Medical Expenditure Survey (1987), where the dependent variable is the number of stays after hospital admission, and where 10 explanatory variables are analysed.     

Bivariate discrete generalized exponential distribution

In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.     

Theoretical results on the discrete Weibull distribution of Nakagawa and Osaki

In this paper, we discuss some theoretical results and properties of the discrete Weibull distribution, which was introduced by Nakagawa and Osaki [The discrete Weibull distribution. IEEE Trans Reliab. 1975;24:300–301]. We study the monotonicity of the probability mass, survival and hazard functions. Moreover, reliability, moments, p-quantiles, entropies and order statistics are also studied. We consider likelihood-based methods to estimate the model parameters based on complete and censored samples, and to derive confidence intervals. We also consider two additional methods to estimate the model parameters. The uniqueness of the maximum likelihood estimate of one of the parameters that index the discrete Weibull model is discussed. Numerical evaluation of the considered model is performed by Monte Carlo simulations. For illustrative purposes, two real data sets are analyzed.     

A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

Summary.  A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.     

A new four-parameter lifetime distribution

A new four-parameter distribution is introduced. It appears to be a distribution allowing for and only allowing for monotonically increasing, bathtub-shaped and upside down bathtub-shaped hazard rates. It contains as particular cases many of the known lifetime distributions. Some mathematical properties of the new distribution, including estimation procedures by the method of maximum likelihood are derived. Some simulations are run to assess the performance of the maximum-likelihood estimators. Finally, the flexibility of the new distribution is illustrated using a real data set.     

Empirical-distribution-function goodness-of-fit tests for discrete models

We present a simple framework for studying empirical-distribution-function goodness-of-fit tests for discrete models. A key tool is a weak-convergence result for an estimated discrete empirical process, regarded as a random element in some suitable sequence space. Special emphasis is given to the problem of testing for a Poisson model and for the geometric distribution. Simulations show that parametric bootstrap versions of the tests maintain a nominal level of significance very closely even for small samples where reliance upon asymptotic critical values is doubtful.     

A new lifetime distribution

Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures, univariate generalizations and bivariate generalizations are derived. A real data application is described to show its superior performance versus at least that of 15 of the known lifetime models.     

On some models leading to the generalized poisson distribution

A new generalization of the Poisson distribution was given by Consul and Jain (1970, 73). Since then more than twenty papers, written by various researchers, have appeared on this model under the titles of Generalized Poisson Distribution (GPD), Lagrangian Poisson distribution or modified power series distribution. Here the author provides two physical models, based on differential-difference equations, which lead to the GPD. A number of axioms are given for a steady state point process which produce the generalized Poisson process. Also, the GPD is derived as the limiting distribution of the two quasi-binomial distributions based on urn models.     

Heine-euler extensions of the poisson distribution

The paper shows that the Heine and Euler distributions (Benkherouf and Bather, 1988) are members of a family of q-series anologues of the Poisson distribution, with similar probability mass functions, but different restrictions on their parameters, and different modes of genesis and properties. The relationships between the Heine, Euler, pseudo-Euler, Poisson and geometric distributions are explored. Illustrative data sets are discussed.     

The probability function of a geometric Poisson distribution

The geometric Poisson (also called Pólya–Aeppli) distribution is a particular case of the compound Poisson distribution. In this study, the explicit probability function of the geometric Poisson distribution is derived and a straightforward proof for this function is given. By means of a proposed algorithm, some numerical examples and an application on traffic accidents are also given to illustrate the usage of the probability function and proposed algorithm.     

The Touchard distribution

We present a novel model, which is a two-parameter extension of the Poisson distribution. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. In contrast to some other generalizations, the Hessian matrix for maximum likelihood estimation of the Touchard parameters has a simple form. We exemplify with three data sets, showing that our suggested model is a competitive candidate for fitting non-Poisson counts.     

The exponential Poisson logarithmic distribution

Abstract

In this article, we proposed a new three parameter lifetime distribution motivated mainly by lifetime issues, which generalizes the Exponential Poisson distribution proposed by Cancho et al. (2011) Cancho, V.G., Louzada-Neto, F., Barriga, G.D. (2011). The poisson-exponential lifetime distribution. Computat. Statist. Data Anal. 55:677–686.[Crossref], [Web of Science ®] , [Google Scholar]. We derive various standard mathematical properties of the proposed model including a formal proof of its probability density function and hazard rate function. The inference via the maximum likelihood approach is discussed. The performance of the maximum likelihood estimators, the likelihood ratio test and its power are studied by simulation. Finally, the proposed model is fitted to two real data sets and it is compared with several models.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

Linear models with an inverse gaussian poisson error distribution

The inverse Gaussian-Poisson (two-parameter Sichel) distribution is useful in fitting overdispersed count data. We consider linear models on the mean of a response variable, where the response is in the form of counts exhibiting extra-Poisson variation, and assume an IGP error distribution. We show how maximum likelihood estimation may be carried out using iterative Newton-Raphson IRLS fitting, where GLIM is used for the IRLS part of the maximization. Approximate likelihood ratio tests are given.