Bivariate Discrete Distributions Research Articles

Visualizing a bivariate discrete distribution and other distributions derived from it

AbstractA single discrete random variable is depicted by a stick diagram, a 2D picture. Naturally, to visualize a bivariate discrete distribution, one can use a bivariate stick diagram, a 3D picture. Unfortunately, many students have difficulty understanding and processing 3D pictures. Therefore, we construct an alternative 2D disc plot to depict the bivariate distribution of (, ), from which we obtain graphically the conditional distributions of given , and given ; the marginal distributions of , , , . Furthermore, we depict the mean and the standard deviation of each distribution using a single‐headed arrow. We hope these visualizations will help students better comprehend these concepts and avoid some misconceptions.

Bayesian geostatistical modeling for discrete‐valued processes

AbstractWe introduce a flexible and scalable class of Bayesian geostatistical models for discrete data, based on nearest‐neighbor mixture processes (NNMP), referred to as discrete NNMP. To define the joint probability mass function (pmf) over a set of spatial locations, we build from local mixtures of conditional pmfs using a directed graphical model, with a directed acyclic graph that summarizes the nearest neighbor structure. The approach supports direct, flexible modeling for multivariate dependence through specification of general bivariate discrete distributions that define the conditional pmfs. In particular, we develop a modeling and inferential framework for copula‐based NNMPs that can attain flexible dependence structures, motivating the use of bivariate copula families for spatial processes. Moreover, the framework allows for construction of models given a pre‐specified family of marginal distributions that can vary in space, facilitating covariate inclusion. Compared to the traditional class of spatial generalized linear mixed models, where spatial dependence is introduced through a transformation of response means, our process‐based modeling approach provides both computational and inferential advantages. We illustrate the methodology with synthetic data examples and an analysis of North American Breeding Bird Survey data.

Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications

In this article, a new family of bivariate discrete distributions is proposed based on the copula concept, in the so-called bivariate discrete odd generalized exponential-G family. Some distributional properties, including the joint probability mass function, joint survival function, joint failure rate function, median correlation coefficient, and conditional expectation, are derived. After proposing the general class, one special model of the new bivariate family is discussed in detail. The maximum likelihood approach is utilized to estimate the family parameters. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood estimators. Finally, the importance of the new bivariate family is explained by means of two distinctive real data sets in various fields.

Regression methods for the appearances of extremes in climate data

AbstractFor any given city, on any calendar day, there will be record high and low temperatures. Which record occurred earlier? If there is a trend towards warming then, intuitively, there should be a preponderance of record highs occurring more recently than the record lows for each of the 365 calendar days. We are interested in modeling the joint distribution of appearances of the extremes but not these values themselves. We develop a bivariate discrete distribution modeling the joint indices of maximum and minimum in a sequence of independent random variables sampled from different distributions. We assume these distributions share a proportional hazard rate and develop regression methods for these paired values. This approach has reasonable power to detect a small mean change over a decade. Using readily available public data, we examine the daily calendar extreme values of five US cities for the decade 2011–2020. We develop linear regression models for these data, describe models to account for calendar‐date dependence, and use diagnostic measures to detect remarkable observations.

On some bivariate discrete distributions with multivariate components

Consider a random vector (X,Y) where X=(X1,X2, ...., Xs,) and Y=(Y1,Y2, ...., Ys) with Xi, Yi, i=1,2,..., s independent non-negative, integer-valued random variables with finite support and such that X>=Y. We show that in the case where the distribution of (YlX=n) is of a certain structural form then there exists a relationship between the distributions of Y and of Y|(X=Y) which uniquely determines the distribution of X. The relationship in question is less stringent that the condition of independence between Y and X-Y usually involved in pro¬blems of this nature. Examples are given to illustrate the result. The case where X,Y have infinite support has been examined earlier by the author.

Inducing a desired value of correlation between two point-scale variables: a two-step procedure using copulas

Focusing on point-scale random variables, i.e. variables whose support consists of the first m positive integers, we discuss how to build a joint distribution with pre-specified marginal distributions and Pearson’s correlation rho . After recalling how the desired value rho is not free to vary between -1 and +1, but generally ranges a narrower interval, whose bounds depend on the two marginal distributions, we devise a procedure that first identifies a class of joint distributions, based on a parametric family of copulas, having the desired margins, and then adjusts the copula parameter in order to match the desired correlation. The proposed methodology addresses a need which often arises when assessing the performance and robustness of some new statistical technique, i.e. trying to build a huge number of replicates of a given dataset, which satisfy—on average—some of its features (for example, the empirical marginal distributions and the pairwise linear correlations). The proposal shows several advantages, such as—among others—allowing for dependence structures other than the Gaussian and being able to accommodate the copula parameter up to an assigned level of precision for rho with a very small computational cost. Based on this procedure, we also suggest a two-step estimation technique for copula-based bivariate discrete distributions, which can be used as an alternative to full and two-step maximum likelihood estimation. Numerical illustration and empirical evidence are provided through some examples and a Monte Carlo simulation study, involving the CUB distribution and three different copulas; an application to real data is also discussed.

Discrete line integral on uniform grids: Probabilistic interpretation and applications

Following the methodology developed by (Comput. Math. Appl. 33 (1997) 81–104), we define a discrete version of gradient vector and associated line integral along arbitrary path connecting two nodes of uniform grid. An exponential representation of joint survival function of bivariate discrete non-negative integer-valued random variables in terms of discrete line integral is established. We apply it to generate a discrete analogue of the Sibuya-type aging property, incorporating many classical and new bivariate discrete models. Several characterizations and closure properties of this class of bivariate discrete distributions are presented.

A bivariate discrete inverse resilience family of distributions with resilience marginals

ABSTRACT In this paper, a new bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distributions, is studied. It is observed that the proposed bivariate distribution is a flexible distribution whose cumulative distribution function has an analytical structure. In addition, a new bivariate geometric distribution can be obtained as a special case. We study different properties of this distribution and propose estimation of its parameters. We will see that the maximum of the variables involved in the proposed bivariate distribution defines some new classes of univariate discrete distributions, which are interesting in their own sake, and can be used to analyze some Reliability systems whose components are positive dependent. Some important futures of this new univariate family of discrete distributions are also studied in details. In addition, a general class of bivariate discrete distributions, whose marginals are exponentiated discrete distributions, is introduced. Moreover, the analysis of two real bivariate data sets is performed to indicate the effectiveness of the proposed models. Finally, we conclude the paper.

A new flexible bivariate discrete Rayleigh distribution generated by the Marshall-Olkin family

In the statistical analysis of bivariate data, it is possible to have discrete observations instead of continuous data, as observed in many studies with survival data. In this study it is introduced a new bivariate discrete distribution derived from two Rayleigh distributions using a method propose d by Marshall and Olkin (1997) where an additional parameter is is introduced to a family of distributions related to the dependence structure of two discrete random variables X1 and X2. The study results show that this new bivariate distribution has good statistical properties and a simple mathematical expression for its correlation coefficient. The usual classical and Bayesian estimators for the parameters of the new distribution are also presented. A simulation study is carried out in order to evaluate some frequentist properties of the proposed model. The usefulness of the proposed model is illustrated using a real medical dataset introduced in the literature in presence of censoring and covariates.

Bayesian and non-Bayesian estimation of four-parameter of bivariate discrete inverse Weibull distribution with applications to model failure times, football and biological data

In this paper we have considered one model, namely the bivariate discrete inverse Weibull distribution, which has not been considered in the statistical literature yet. The proposed model is a discrete analogue of Marshall-Olkin inverse Weibull distribution. Some of its important statistical properties are studied. Maximum likelihood and Bayesian methods are used to estimate the model parameters. A detailed simulation study is carried out to examine the bias and mean square error of maximum likelihood and Bayesian estimators. Finally, three real data sets are analyzed to illustrate the importance of the proposedmodel.

Discrete Weibull regression for modelling football outcomes

We propose the use of the discrete Weibull distribution for modelling football match results, as an alternative to existing Poisson and generalised Poisson models. The number of goals scored by the two teams playing a football match are regarded as a pairwise observation and are modelled first through two independent discrete Weibull variables, and then through two dependent discrete Weibull variables, using a copula approach that accommodates non-null correlation. The parameters of the bivariate discrete Weibull distributions are assumed to depend on covariates such as the attack and defence abilities of the two teams and the 'home effect'. Several discrete Weibull regression models are proposed and then applied to the 2015-2016 Italian Serie A. Even if the interpretation of parameters is less immediate than in the case of bivariate Poisson models, nevertheless these models represent a suitable alternative, which can be applied also in other fields than sport data analysis.

Bivariate exponentiated discrete Weibull distribution: statistical properties, estimation, simulation and applications

In this paper, a new bivariate discrete distribution is defined and studied in-detail, in the so-called the bivariate exponentiated discrete Weibull distribution. Several of its statistical properties including the joint cumulative distribution function, joint probability mass function, joint hazard rate function, joint moment generating function, mathematical expectation and reliability function for stress–strength model are derived. Its marginals are exponentiated discrete Weibull distributions. Hence, these marginals can be used to analyze the hazard rates in the discrete cases. The model parameters are estimated using the maximum likelihood method. Simulation study is performed to discuss the bias and mean square error of the estimators.Finally, two real data sets are analyzed to illustrate the flexibility of the proposed model.

Discrete analogues of continuous bivariate probability distributions

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.

On a General Class of Discrete Bivariate Distributions

In this paper we develop a general class of bivariate discrete distributions. The basic idea is quite simple. The marginals are obtained by taking the random geometric sum of the baseline random variables. The proposed class of distributions is a flexible class of bivariate discrete distributions in the sense the marginals can take variety of shapes. The probability mass functions of the marginals can be heavy tailed, unimodal as well as multimodal. It can be both over dispersed as well as under dispersed. We discuss different properties of the proposed class of bivariate distributions. The proposed distribution has some interesting physical interpretations also. Further, we consider two specific base line distributions: Poisson and negative binomial distributions for illustrative purposes. Both of them are infinitely divisible. The maximum likelihood estimators of the unknown parameters cannot be obtained in closed form. They can be obtained by solving three and five dimensional non-linear optimizations problems, respectively. To avoid that we propose to use expectation maximization algorithm, and it is observed that the proposed algorithm can be implemented quite easily in practice. We have performed some simulation experiments to see how the proposed EM algorithm performs, and it works quite well in both the cases. The analysis of one real data set has been performed to show the effectiveness of the proposed class of models. Finally, we discuss some open problems and conclude the paper.

On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.

A bivariate count model with discrete Weibull margins

Multivariate discrete data arise in many fields (statistical quality control, epidemiology, failure and reliability analysis, etc.) and modelling such data is a relevant task. Here we consider the construction of a bivariate model with discrete Weibull margins, based on Farlie–Gumbel–Morgenstern copula, analyse its properties especially in terms of attainable correlation, and propose several methods for the point estimation of its parameters. Two of them are the standard one-step and two-step maximum likelihood procedures; the other two are based on an approximate method of moments and on the method of proportion, which represent intuitive alternatives for estimating the dependence parameter. A Monte Carlo simulation study is presented, comprising more than one hundred artificial settings, which empirically assesses the performance of the different estimation techniques in terms of statistical properties and computational cost. For illustrative purposes, the model and related inferential procedures are fitted and applied to two datasets taken from the literature, concerning failure data, presenting either positive or negative correlation between the two observed variables. The applications show that the proposed bivariate discrete Weibull distribution can model correlated counts even better than existing and well-established joint distributions.

Some results about bivariate discrete distributions through the vector of aging intensities

ABSTRACTIn this paper bivariate vectors of discrete aging and alternative aging intensities are introduced. Using these vector-valued functions we present some results about bivariate discrete distributions.

Discrete Weibull regression for modeling football outcomes

We propose the use of the discrete Weibull distribution for modelling football match results, as an alternative to existing Poisson and generalised Poisson models. The number of goals scored by the two teams playing a football match are regarded as a pairwise observation and are modelled first through two independent discrete Weibull variables, and then through two dependent discrete Weibull variables, using a copula approach that accommodates non-null correlation. The parameters of the bivariate discrete Weibull distributions are assumed to depend on covariates such as the attack and defence abilities of the two teams and the 'home effect'. Several discrete Weibull regression models are proposed and then applied to the 2015-2016 Italian Serie A. Even if the interpretation of parameters is less immediate than in the case of bivariate Poisson models, nevertheless these models represent a suitable alternative, which can be applied also in other fields than sport data analysis.

On some distributions arising from a generalized trivariate reduction scheme

In this article we construct bivariate discrete distributions in Z2. We make use of a generalized trivariate reduction technique. The special case leading to a generalization of a bivariate Skellam distribution is studied in detail. Properties of the derived models as well as estimation are examined. Real data application is provided. Discussion of extensions to different models is also mentioned.

On a Simple Construction of a Bivariate Probability Function With a Common Marginal

We introduce a family of bivariate discrete distributions whose members are generated by a decreasing mass function p, and with margins given by p. Several properties and examples are obtained, including a family of seemingly novel bivariate Poisson distributions.