Generating discrete analogues of continuous probability distributions-A survey of methods and constructions

Chakraborty, Subrata

doi:10.1186/s40488-015-0028-6

Cited by 117 publications

(109 citation statements)

References 83 publications

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“…Several authors including Kemp (), Inusah and Kozubowski (), Kozubowski and Inusah () have discretized a continuous random variable X with PDF f ( x ) by using the PMF of the form

P ((), Y = y) = f ((), y) / \sum_{t = - \infty}^{\infty} f ((), t), y = 0, \pm 1, \pm 2, \dots,

when the support of X is the whole real line. If the support of X is [0, ∞), then y = 0, 1, 2, … Chakraborty () named the method in Equation as Methodology II.…”

Section: Methods For Generating Univariate Discrete Distributionsmentioning

confidence: 99%

“…where S is the survival function of a continuous random variable Y with support [0, ∞). Chakraborty (2015) called the technique in Equation (10) Methodology IV. Nekoukhou et al (2013) used the form in Equation (10), where the continuous random variable Y has an EED and obtained the CDF of DGE distribution of the second type (denoted by DGE 2 (c, p)) as F y ð Þ = 1−p y + 1 À Á c , y = 0,1,2,3,…where 0 < p = e − λ < 1 and c > 0:…”

Section: Discretizing a Continuous Random Variable: Using A Survivamentioning

confidence: 99%

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Review of univariate and bivariate exponentiated exponential‐geometric distributions

Famoye

Lee

2019

WIREs Computational Stats

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The exponentiated exponential (or generalized exponential) distribution is a two‐parameter right skewed unimodal density function. A discrete analogue of the distribution is often called the exponentiated exponential‐geometric (or discrete generalized exponential) distribution. We review some methods for generating discrete analogues of the exponentiated exponential distribution and discuss some of their properties. Among the important properties of the univariate distribution are (a) closed form cumulative distribution function and probability mass function and (b) ability to accommodate under‐, equi‐, or over‐dispersion. Some methods to extend the univariate distribution to the bivariate exponentiated exponential‐geometric distribution are discussed. The extension of univariate and bivariate distributions to count data regression models is briefly described. It is expected that this review will serve as a reference and encourage further research work and generalizations of the exponentiated exponential‐geometric distribution. This article is categorized under: Statistical Models > Linear Models Data: Types and Structure > Categorical Data Statistical Models > Generalized Linear Models

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“…Several authors including Kemp (), Inusah and Kozubowski (), Kozubowski and Inusah () have discretized a continuous random variable X with PDF f ( x ) by using the PMF of the form

P ((), Y = y) = f ((), y) / \sum_{t = - \infty}^{\infty} f ((), t), y = 0, \pm 1, \pm 2, \dots,

when the support of X is the whole real line. If the support of X is [0, ∞), then y = 0, 1, 2, … Chakraborty () named the method in Equation as Methodology II.…”

Section: Methods For Generating Univariate Discrete Distributionsmentioning

confidence: 99%

Section: Discretizing a Continuous Random Variable: Using A Survivamentioning

confidence: 99%

Review of univariate and bivariate exponentiated exponential‐geometric distributions

Famoye

Lee

2019

WIREs Computational Stats

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“…There is a natural link between the DW distribution and the continuous Weibull distribution with interval‐censored data. The DW distribution was in fact developed as a discretized form of the continuous Weibull distribution (Chakraborty, ). In particular, let Y be a random variable distributed as a continuous Weibull distribution, with probability density function and cumulative density function given by

\begin{matrix} f_{normalW} false(y; q, β false) = β log false(q false) y^{(β - 1)} exp false{- y^{β} log (q) false} y \geq 0, \\ F_{normalW} false(y; q, β false) = 1 - exp false{- y^{β} log (q) false} \end{matrix}

respectively.…”

Section: Discrete Weibull Distributionmentioning

confidence: 99%

Discrete Weibull Generalized Additive Model: An Application to Count Fertility Data

Peluso

Vinciotti

2018

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary Fertility plans, measured by the number of planned children, have been found to be affected by education and family background via complex tail dependences. This challenge was previously met with the use of non‐parametric jittering approaches. The paper shows how a novel generalized additive model based on a discrete Weibull distribution provides partial effects of the covariates on fertility plans which are comparable with jittering, without the inherent drawback of conditional quantiles crossing. The model has some additional desirable features: both overdispersed and underdispersed data can be modelled by this distribution, the conditional quantiles have a simple analytic form and the likelihood is the same as that of a continuous Weibull distribution with interval‐censored data. Because the likelihood is like that of a continuous Weibull distribution, efficient implementations are already available, in the R package gamlss, for a range of models and inferential procedures, and at a fraction of the time compared with the jittering and Conway–Maxwell–Poisson approaches, showing potential for the wide applicability of this approach to the modelling of count data.

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“…Some properties for a discrete analogue to continuous distributions obtained by this method were studied by Kemp (2004), Bracquemond and Gaudoin (2003), , Chakraborty (2015), among others.…”

Section: Discretization By Survival Functionmentioning

confidence: 99%

“…In recent years, the generation of a discrete observation from a continuous random variable has been considered by several authors (see, for example, Chakraborty (2015)). Basically, the main purpose to discretize a continuous probability density function is to generate a distribution for the analysis of strictly discrete data.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study Between Two Discrete Lindley Distributions

Oliveira

Mazucheli

Achcar

2017

CeN

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The methods to generate a probability function from a probability density function has long been used in recent years. In general, the discretization process produces probability functions that could be alternatives to traditional distributions used in the analysis of count data as geometric, Poisson and negative binomial distributions. The discretization also avoids the use of a continuous distribution in the analysis of strictly discrete data. In this paper, using the method based on an infinite series, proposed by , we studied an alternative discrete Lindley distribution to those studies in Bakouch et al. (2014). For both distributions, a simulation study is carried out to examine the bias and mean squared error for the maximum likelihood estimators of the parameters as well the coverage probability and the length of coverage probability. For the discrete Lindley distribution obtained by infinite series method, we present the analytical expression for bias reduction of the maximum likelihood estimator. Some examples using real data from the literature show the potential of these distributions. Despite the discretization methods are quite different, the resulting distributions are interchangeable, however the distribution generated by an infinite series has simple mathematical expressions and can be used directly to count data in the presence of covariates.

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Generating discrete analogues of continuous probability distributions-A survey of methods and constructions

Cited by 117 publications

References 83 publications

Review of univariate and bivariate exponentiated exponential‐geometric distributions

Review of univariate and bivariate exponentiated exponential‐geometric distributions

Discrete Weibull Generalized Additive Model: An Application to Count Fertility Data

A Comparative Study Between Two Discrete Lindley Distributions

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