Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.     

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.     

Multivariate log-skewed distributions with normal kernel and their applications

We introduce two classes of multivariate log-skewed distributions with normal kernel: the log canonical fundamental skew-normal (log-CFUSN) and the log unified skew-normal. We also discuss some properties of the log-CFUSN family of distributions. These new classes of log-skewed distributions include the log-normal and multivariate log-skew normal families as particular cases. We discuss some issues related to Bayesian inference in the log-CFUSN family of distributions, mainly we focus on how to model the prior uncertainty about the skewing parameter. Based on the stochastic representation of the log-CFUSN family, we propose a data augmentation strategy for sampling from the posterior distributions. This proposed family is used to analyse the US national monthly precipitation data. We conclude that a high-dimensional skewing function lead to a better model fit.     

Stochastic ordering and a class of multivariate new better than used distributions

A new characterization for the univariate class of new better than used ‘NBU’ distributions in terms of stochastic ordering is introduced. A multivariate version of this characterization is then used to define a multivariate class of NBU distributions. Basic properties of this class are derived. Comparisons and relationships of this new class with earlier classes are developed. Two multivariate new worse than used (NWU) classes of life distributions are defined and compared and their basic properties are studied.     

On the monotonic properties of discrete failure rates

As is well known, the monotonicity of failure rate of a life distribution plays an important role in modeling failure time data. In this paper, we develop techniques for the determination of increasing failure rate (IFR) and decreasing failure rate (DFR) property for a wide class of discrete distributions. Instead of using the failure rate, we make use of the ratio of two consecutive probabilities. The method developed is applied to various well known families of discrete distributions which include the binomial, negative binomial and Poisson distributions as special cases. Finally, a formula is presented to determine explicitly the failure rate of the families considered. This formula is used to determine the failure rate of various classes of discrete distributions. These formulas are explicit but complicated and cannot normally be used to determine the monotonicity of the failure rates.     

On describing multivariate skewed distributions: A directional approach

Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. These measures were designed for testing the hypothesis of distributional symmetry; their relevance for describing skewed distributions is less obvious. In this article, the authors consider the problem of characterizing the skewness of multivariate distributions. They define directional skewness as the skewness along a direction and analyze two parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of classes of distributions for particular applications. The authors use the concept of directional skewness twice in the context of Bayesian linear regression under skewed error: first in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.     

Modeling the Agreement of Discrete Bivariate Survival Times using Kappa Coefficient

Using local kappa coefficients, we develop a method to assess the agreement between two discrete survival times that are measured on the same subject by different raters or methods. We model the marginal distributions for the two event times and local kappa coefficients in terms of covariates. An estimating equation is used for modeling the marginal distributions and a pseudo-likelihood procedure is used to estimate the parameters in the kappa model. The performance of the estimation procedure is examined through simulations. The proposed method can be extended to multivariate discrete survival distributions.     

Testing for departures from nominal dispersion in generalized nonlinear models with varying dispersion and/or additive random effects

This paper discusses the tests for departures from nominal dispersion in the framework of generalized nonlinear models with varying dispersion and/or additive random effects. We consider two classes of exponential family distributions. The first is discrete exponential family distributions, such as Poisson, binomial, and negative binomial distributions. The second is continuous exponential family distributions, such as normal, gamma, and inverse Gaussian distributions. Correspondingly, we develop a unifying approach and propose several tests for testing for departures from nominal dispersion in two classes of generalized nonlinear models. The score test statistics are constructed and expressed in simple, easy to use, matrix formulas, so that the tests can easily be implemented using existing statistical software. The properties of test statistics are investigated through Monte Carlo simulations.     

Geeta distribution and its properties

A new discrete distribution defined over all the positive integers and with the name of Geeta distribution is described. It is L-shaped like the logarithmic series distribution, Yule distribution and the discrete Pareto distribution but is far more versatile than them as it has two parameters. It belongs to the classes of location parameter distributions, modified power series distributions, Lagrange series distributions and exponential distributions. Its mean fi, variance a2 and two recurrence formulae for higher central moments are obtained. Convolution theorem and variations in the model with changes in the parameters have been considered. ML estimators, MVU estimators and estimators based of mean and variance and on mean and first frequency have been derived.     

Global prior distributions for the analysis of discrete graphical models

Summary We propose a new class of prior distributions for the analysis of discrete graphical models. Such a class, obtained following a conditional approach, generalizes the hyper Dirichlet distributions of Dawid and Lauritzen (1993), since it can be extended to non decomposable graphical models. The two classes are compared in terms of model selection, with an application to a medical data-set illustrating the performance of the two resulting procedures. The proposed class turns out to select simpler, more par-simonious structures.     

Bayesian multivariate Poisson mixtures with an unknown number of components

In this paper we present Bayesian analysis of finite mixtures of multivariate Poisson distributions with an unknown number of components. The multivariate Poisson distribution can be regarded as the discrete counterpart of the multivariate normal distribution, which is suitable for modelling multivariate count data. Mixtures of multivariate Poisson distributions allow for overdispersion and for negative correlations between variables. To perform Bayesian analysis of these models we adopt a reversible jump Markov chain Monte Carlo (MCMC) algorithm with birth and death moves for updating the number of components. We present results obtained from applying our modelling approach to simulated and real data. Furthermore, we apply our approach to a problem in multivariate disease mapping, namely joint modelling of diseases with correlated counts.     

Discrete random vectors generated by Taylor expansions

We demonstrate how univariate discrete and multivariate discrete distributions can be generated using Taylor expansions. Some of the results involve use of Bell polynomials.     

Some properties of discrete bathtub-shaped distributions

The existence and the usefulness of discrete bathtub-shaped and upside down bathtub-shaped distributions have been demonstrated in some papers of recent origin. However, the general properties of these two classes of distributions do not seem to have been discussed. This article proposes to study some reliability properties of such distributions. We investigate the closure properties with reference to convolution, mixing, series and parallel systems, etc. and existence of bounds on reliability functions, moment properties, and convergence.     

Expectation formulas for integer valued multivariate random variables

Nadarajah 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.     

Accounting for uncertainty in extremal dependence modeling using Bayesian model averaging techniques

Modeling the joint tail of an unknown multivariate distribution can be characterized as modeling the tail of each marginal distribution and modeling the dependence structure between the margins. Classical methods for modeling multivariate extremes are based on the class of multivariate extreme value distributions. However, such distributions do not allow for the possibility of dependence at finite levels that vanishes in the limit. Alternative models have been developed that account for this asymptotic independence, but inferential statistical procedures seeking to combine the classes of asymptotically dependent and asymptotically independent models have been of limited use. We overcome these difficulties by employing Bayesian model averaging to account for both types of asymptotic behavior, and for subclasses within the asymptotically independent framework. Our approach also allows for the calculation of posterior probabilities of different classes of models, allowing for direct comparison between them. We demonstrate the use of joint tail models based on our broader methodology using two oceanographic datasets and a brief simulation study.     

CHARACTERIZATIONS OF MULTINOMIAL AND NEGATIVE MULTINOMIAL MIXTURES BY REGRESSION

Characterizations are given for mixtures of multinomial and negative multinomial distributions with respect to their index parameter. Several well known multivariate discrete distributions are used as illustrative examples.     

Setting the clock back to zero property of a class of bivariate life distributions

In a previous paper. B. R. Rao and Talwalker (1993) considered absolutely continuous life distributions and extended the Lack of Memory Property (L.M.P.) of the exponential distribution and showed that several classes of life distributions have this property, which was called the 'setting the clock back to zero' property. ¶Its analog is discussed in the present paper for hivariate and multivariate classes of life distributions. As a simple application of this analog, it is proved that the Life expectancy and the Percentile Residual Life vectors of a population of individuals under the influence of multiple competing risks have simple expressions if the class of their joint life distributions has the setting the clock back to zero property,     

Efficient time/space algorithm to compute rectangular probabilities of multinomial, multivariate hypergeometric and multivariate Pólya distributions

The computation of rectangular probabilities of multivariate discrete integer distributions such as the multinomial, multivariate hypergeometric or multivariate Pólya distributions is of great interest both for statistical applications and for probabilistic modeling purpose. All these distributions are members of a broader family of multivariate discrete integer distributions for which computationaly efficient approximate methods have been proposed for the evaluation of such probabilities, but with no control over their accuracy. Recently, exact algorithms have been proposed for computing such probabilities, but they are either dedicated to a specific distribution or to very specific rectangular probabilities. We propose a new algorithm that allows to perform the computation of arbitrary rectangular probabilities in the most general case. Its accuracy matches or even outperforms the accuracy exact algorithms when the rounding errors are taken into account. In the worst case, its computational cost is the same as the most efficient exact method published so far, and is much lower in many situations of interest. It does not need any additional storage than the one for the parameters of the distribution, which allows to deal with large dimension/large counting parameter applications at no extra memory cost and with an acceptable computation time, which is a major difference with respect to the methods published so far.     

Linear Conditional Expectation for Discretized Distributions

Many statistical methods for continuous distributions assume a linear conditional expectation. Components of multivariate distributions are often measured on a discrete ordinal scale based on a discretization of an underlying continuous latent variable. The results in this paper show that common examples of discretized bivariate and trivariate distributions will have a linear conditional expectation. Examples and simulations are provided to illustrate the results.