Discrete Bivariate Distributions Research Articles

Discrete Bivariate Distributions Generated By Copulas: DBEEW Distribution

Copulas are the most powerful tools in constructing continuous multivariate distributions given marginals and certain dependence structures. In this paper, we shall propose a general and novel method of generating discrete bivariate distributions using copulas. The advantage of our method is that, contrary to the standard methods, we do not need to have the joint distribution of the base variables, but we only need the marginal distributions. In particular, we shall concentrate on generating a new family of discrete bivariate exponentiated extended Weibull (DBEEW) distribution by a Cuadras–Auge copula. We shall study several mathematical properties of a DBEEW distribution. Three special submodels of our new proposed distribution are studied in more detail. Finally, estimation of parameters for these three submodels is investigated and their potential and flexibility are examined using a real data set.

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.

Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach

ABSTRACTThe modeling and analysis of lifetime data in which the main endpoints are the times when an event of interest occurs is of great interest in medical studies. In these studies, it is common that two or more lifetimes associated with the same unit such as the times to deterioration levels or the times to reaction to a treatment in pairs of organs like lungs, kidneys, eyes or ears. In medical applications, it is also possible that a cure rate is present and needed to be modeled with lifetime data with long-term survivors. This paper presented a comparative study under a Bayesian approach among some existing continuous and discrete bivariate distributions such as the bivariate exponential distributions and the bivariate geometric distributions in presence of cure rate, censored data and covariates. In presence of lifetimes related to cured patients, it is assumed standard mixture cure rate models in the data analysis. The posterior summaries of interest are obtained using Markov Chain Monte Carlo methods. To illustrate the proposed methodology two real medical data sets are considered.

Basu-Dhar's bivariate geometric distribution in presence of censored data and covariates: some computational aspects

Some computational aspects to obtain classical and Bayesian inferences for the Basu and Dhar (1995) bivariate geometric distribution in presence of censored data and covariates are discussed in this paper. The posterior summaries of interest are obtained using standard existing MCMC (Markov Chain Monte Carlo) simulation methods available in popular free softwares as the OpenBugs software and the R software. Numerical illustrations are introduced considering simulated and real datasets showing that the use of discrete bivariate distributions may be a good alternative to the use of continuous bivariate distributions, in many areas of application.

Expectation formulas for integer valued multivariate random variables

ABSTRACTNadarajah and Mitov [Communications in Statistics—Theory and Methods, 32, 2003, 47–60] derived an expectation formula for continuous multivariate random variables involving the joint survival function. Their result is extended here for discrete multivariate random variables. Examples proposing new discrete bivariate distributions are given.

A new general class of discrete bivariate distributions constructed by using the likelihood ratio

In statistics, stochastic orders formalize such a concept that one random variable is bigger than another. In this paper, we develop a new class of discrete bivariate distributions based on a stochastic order defined by the likelihood ratio. We derive general formula for the joint distributions belonging to the class. It will be seen that, from the proposed class, specific families of distributions can be efficiently generated just by specifying the ‘baseline seed distributions’. An important feature of the proposed discrete bivariate model is that, unlike other discrete bivariate models already proposed in the literature such as the well-known and most popular bivariate Poisson distribution by Holgate, it can model both positive and negative dependence. A number of new families of discrete bivariate distributions are generated from the proposed class. Furthermore, the generated bivariate distributions are applied to analyze real data sets and the results are compared with those obtained from some conventional models.

Modeling Discrete Bivariate Data with Applications to Failure and Count Data

In this study, we propose a new class of flexible bivariate distributions for discrete random variables. The proposed class of distribution is based on the notion of conditional failure rate for a discrete‐type random variable. We derive general formulae for the joint distributions belonging to the proposed class that, unlike other discrete bivariate models already proposed in the literature such as the well‐known and most popular Holgate's bivariate Poisson distribution, can model both positive and negative dependence. We discuss general statistical properties of the proposed class as well. Specific families of bivariate distributions can be generated from the general class proposed in this paper just by specifying the ‘baseline distributions’. Furthermore, specific discrete bivariate distributions belonging to the proposed class are applied to analyze three real data sets, and the results are compared with those obtained from conventional models. Copyright © 2017 John Wiley & Sons, Ltd.

New classes of discrete bivariate distributions with application to football data

ABSTRACTMotivated by Lee and Cha [The American Statistician 69 (2015) 221–230], we introduce 18 new families of discrete bivariate distributions. We give explicit expressions for their probability mass functions, cumulative distribution functions, product moments, and moment-generating functions. We reanalyze the football data in Lee and Cha (2015) and show that some of the newly proposed distributions provide better fits than the two families proposed by Lee and Cha.

On Two General Classes of Discrete Bivariate Distributions

In this article, we develop two general classes of discrete bivariate distributions. We derive general formulas for the joint distributions belonging to the classes. The obtained formulas for the joint distributions are very general in the sense that new families of distributions can be generated just by specifying the “baseline seed distributions.” The dependence structures of the bivariate distributions belonging to the proposed classes, along with basic statistical properties, are also discussed. New families of discrete bivariate distributions are generated from the classes. Furthermore, to assess the usefulness of the proposed classes, two discrete bivariate distributions generated from the classes are applied to analyze a real dataset and the results are compared with those obtained from conventional models.

A family of composite discrete bivariate distributions with uniform marginals for simulating realistic and challenging optimization-problem instances

We consider a family of composite bivariate distributions, or probability mass functions (pmfs), with uniform marginals for simulating optimization-problem instances. For every possible population correlation, except the extreme values, there are an infinite number of valid joint distributions in this family. We quantify the entropy for all member distributions, including the special cases under independence and both extreme correlations. Greater variety is expected across optimization-problem instances simulated based on a high-entropy pmf. We present a closed-form solution to the problem of finding the joint pmf that maximizes entropy for a specified population correlation, and we show that this entropy-maximizing pmf belongs to our family of pmfs. We introduce the entropy range as a secondary indicator of the variety of instances that may be generated for a particular correlation. Finally, we discuss how to systematically control entropy and correlation to simulate a set of synthetic problem instances that includes challenging examples and examples with realistic characteristics.

The Assessment of Performance of Correlation Estimates in Discrete Bivariate Distributions Using Bootstrap Methodology

Little attention has been given to the correlation coefficient when data come from discrete or continuous non-normal populations. In this article, we consider the efficiency of two correlation coefficients which are from the same family, Pearson's and Spearman's estimators. Two discrete bivariate distributions were examined: the Poisson and the Negative Binomial. The comparison between these two estimators took place using classical and bootstrap techniques for the construction of confidence intervals. Thus, these techniques are also subject to comparison. Simulation studies were also used for the relative efficiency and bias of the two estimators. Pearson's estimator performed slightly better than Spearman's.

On Minimum Divergence Adaptation of Discrete Bivariate Distributions to Given Marginals

This correspondence considers the class P of discrete bivariate distributions p with marginals a(p), b(p) and the subclass Pa,b /spl sub/ P of distributions with given marginals a and b. For an arbitrary q /spl isin/ P we study the distributions p/spl tilde/ /spl isin/ P/sub a,b/ minimizing on P/sub a,b/ various /spl phi/-divergences D/sub /spl phi//(p, q) for convex /spl phi/ with finite /spl phi/(0).

Univariate and Bivariate Loglinear Models for Discrete Test Score Distributions

The well-developed theory of exponential families of distributions is applied to the problem of fitting the univariate histograms and discrete bivariate frequency distributions that often arise in the analysis of test scores. These models are powerful tools for many forms of parametric data smoothing and are particularly well-suited to problems in which there is little or no theory to guide a choice of probability models, e.g., smoothing a distribution to eliminate roughness and zero frequencies in order to equate scores from different tests. Attention is given to efficient computation of the maximum likelihood estimates of the parameters using Newton's Method and to computationally efficient methods for obtaining the asymptotic standard errors of the fitted frequencies and proportions. We discuss tools that can be used to diagnose the quality of the fitted frequencies for both the univariate and the bivariate cases. Five examples, using real data, are used to illustrate the methods of this paper.

UNIVARIATE AND BIVARIATE LOGLINEAR MODELS FOR DISCRETE TEST SCORE DISTRIBUTIONS

ABSTRACTThe well‐developed theory of exponential families of distributions is applied to the problem of fitting the univariate histograms and discrete bivariate frequency distributions that often arise in the analysis of test scores. These models are powerful tools for many forms of parametric data smoothing and are particularly well‐suited to problems where there is little or no theory to guide a choice of probability models, e.g. smoothing a distribution to eliminate roughness and zero frequencies in order to equate scores from different tests. Attention is given to efficient computation of the maximum likelihood estimates of the parameters using Newton's Method and to computationally efficient methods for obtaining the asymptotic standard errors of the fitted frequencies and proportions. We discuss tools that can be used to diagnose the quality of the fitted frequencies for both the univariate and the bivariate cases. Five examples, using real data, are used to illustrate the methods of this paper.

Discrete bivariate distributions with given marginals and correlation

Using convex linear combinations of the probability functions for the discrete Frechet boundary distributions and the probability function for independent random variables, we construct bivariate probability functions for dependent discrete random variables with arbitrary marginals and any correlation between the theoretical minimum and maximum. The technique is elementary and the results can be adapted readily for simulation.

Inequalities for Distributions with Given Marginals

An ordering on discrete bivariate distributions formalizing the notion of concordance is defined and shown to be equivalent to stochastic ordering of distribution functions with identical marginals. Furthermore, for this ordering, $\int\varphi dH$ is shown to be $H$-monotone for all superadditive functions $\varphi$, generalizing earlier results of Hoeffding, Frechet, Lehmann and others. The usual correlation coefficient, Kendall's $\tau$ and Spearman's $\rho$ are shown to be monotone functions of $H$. That $\int\varphi dH$ is $H$-monotone holds for distributions on $\mathbb{R}^n$ with fixed $(n - 1)$-dimensional marginals for any $\varphi$ with nonnegative finite differences of order $n$. Some related results are obtained. Stochastic ordering is preserved under certain transformations, e.g., convolutions. A distribution on $\mathbb{R}^\infty$ is constructed, making $\max(X_1,\cdots, X_n)$ stochastically largest for all $n$ when $X_i$ have given one-dimensional distributions, generalizing a result of Robbins. Finally an ordering for doubly stochastic matrices is proposed.