A generalization of the bivariate Beta-Binomial distribution

This paper presents a new bivariate discrete distribution that generalizes the bivariate Beta-Binomial distribution. It is generated by Appell hypergeometric function F₁ and can be obtained as a Binomial mixture with an Exton's Generalized Beta distribution. The model has different marginal distributions which are, together with the conditional distributions, more flexible than the Beta-Binomial distribution. It has non-linear regression curves and is useful for random variables with positive correlation. These features make the model very adequate to fit observed data as the two applications included show.     

Sums, products and ratios of generalized beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products and ratios of independent beta random variables. In this paper, we extend professor Pham-Gia's results when X ₁ and X ₂ are independent random variables distributed according to two generalized beta distributions. For each of these distributions, we derive exact expressions for the densities of S=X ₁+X ₂, D=X ₁−X ₂, P=X ₁ X ₂, and R=X ₂/X ₁. The expressions turn out to involve the incomplete beta function as well as the hypergeometric functions of one and two variables.     

Minimal Markov Basis for Tests of Main Effect Models for 2p-1 Fractional Factorial Designs of Resolution p

We consider conditional exact tests of factor effects in design of experiments for discrete response variables. Similarly to the analysis of contingency tables, Markov chain Monte Carlo methods can be used to perform exact tests, especially when large-sample approximations of the null distributions are poor and the enumeration of the conditional sample space is infeasible. In order to construct a connected Markov chain over the appropriate sample space, one approach is to compute a Markov basis. Theoretically, a Markov basis can be characterized as a generator of a well-specified toric ideal in a polynomial ring and is computed by computational algebraic software. However, the computation of a Markov basis sometimes becomes infeasible, even for problems of moderate sizes. In the present article, we obtain the closed-form expression of minimal Markov bases for the main effect models of 2^{p ? 1} fractional factorial designs of resolution p.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Discrete generalized exponential distribution of a second type

In this paper, we shall attempt to introduce another discrete analogue of the generalized exponential distribution of Gupta and Kundu [Generalized exponential distributions, Aust. N. Z. J. Stat. 41(2) (1999), pp. 173–188], different to that of Nekoukhou et al. [A discrete analogue of the generalized exponential distribution, Comm. Stat. Theory Methods, to appear (2011)]. This new discrete distribution, which we shall call a discrete generalized exponential distribution of the second type (DGE₂(α, p)), can be viewed as another generalization of the geometric distribution. We shall first study some basic distributional and moment properties, as well as order statistics distributions of this family of new distributions. Certain compounded DGE₂(α, p) distributions are also discussed as the results of which some previous lifetime distributions such as that of Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42] follow as corollaries. Then, we will investigate estimation of the parameters involved. Finally, we will examine the model with a real data set.     

A generalization for two-sided power distributions and adjusted method of moments

A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a new class of discrete distributions on {0, 1, …, k}. In addition, a new numerical method for parameter estimation using moments is given.     

Discrete distributions with maximum expected value of the maximum

We give an upper bound for the expected value of the largest order statistic of a simple random sample of size n from a discrete distribution on N points. We also characterize the distributions that attain such bound. In the particular case n=2, we obtain a characterization of the discrete uniform distribution. © 1998 Elsevier Science B.V. All rights reserved.     

On the problem of the unique characterization of discrete distributions by single regression of weak records

We consider the problem of the unique identification of discrete probability distributions by the single regression function of non-adjacent discrete weak record values. We present a new approach to this problem and we show that the uniqueness of the characterization is equivalent to the uniqueness of solution to a corresponding difference equation or an appropriate system of difference equations. This result is applied to obtain known as well as new characterizations of discrete distributions.     

A review of the CTP distribution: a comparison with other over- and underdispersed count data models

The complex triparametric Pearson (CTP) distribution is a flexible model belonging to the Gaussian hypergeometric family that can account for over- and underdispersion. However, despite its good properties, not much attention has been paid to it. So, we revive the CTP comparing it with some well-known distributions that cope with overdispersion (negative binomial, generalized Poisson and univariate generalized Waring) as well as underdispersion (Conway–Maxwell–Poisson (CMP) and hyper-Poisson (HP)). We make a simulation study that reveals the performance of the CTP and shows that it has its own space among count data models. In this sense, we also explore some overdispersed datasets which seem to be more appropriately modelled by the CTP than by other usual models. Moreover, we include two underdispersed examples to illustrate that the CTP can provide similar fits to the CMP or HP (sometimes even more accurate) without the computational problems of these models.     

A note on exact calculation of the non central hypergeometric distribution

Direct calculation of the non central hypergeometric (NH) distribution and its moments can present computational issues in both efficiency and accuracy. In response, several methods, both approximate and exact, for calculating the NH mean and variance have appeared in the literature. We add to this body of work, a straight-forward, exact method that is easily programed, efficient, and computationally stable. Specifically, by considering the logs of the values of the NH probability mass function (pmf) and then shifting the exponents so that, prior to normalization, the mode acquires a value of 1, concerns for overflow are eliminated.     

Limiting distributions of MLE and UMVUE in the biparametric uniform distribution

In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ₁, θ₂) when sampling from a biparametric uniform distributionU(θ₁, θ₂). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution of UMVUE is a shift of the limiting distribution of MLE.     

Prediction of Variables Via Their Order Statistics in Bivariate Elliptical Distributions with Application in the Financial Markets

Assuming that (X₁, X₂) has a bivariate elliptical distribution, we obtain an exact expression for the joint probability density function (pdf) as well as the corresponding conditional pdfs of X₁ and X₍₂₎ ? max?{X₁, X₂}. The problem is motivated by an application in financial markets. Exchangeable random variables are discussed in more detail. Two special cases of the elliptical distributions that is the normal and the student’s t models are investigated. For illustrative purposes, a real data set on the total personal income in California and New York is analyzed using the results obtained. Finally, some concluding remarks and further works are discussed.     

On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

Inference Under Right Censoring in a Discrete Setup

In this article, we consider the right random censoring scheme in a discrete setup when the lifetime and censoring variables are independent and have geometric distributions with means 1/θ₁ and 1/θ₂, respectively. We first obtain the Maximum Likelihood and Method of Moment estimators of the unknown parameters. We also find the Bayes and Posterior Regret Gamma Minimax estimators of the parameters for the two cases when the prior distributions are dependent and independent, assuming a squared error loss function. We then discuss the Proportional Hazard model, and obtain Maximum Likelihood estimators of the unknown parameters and derive the Bayes estimators assuming squared error loss using Markov Chain Monte Carlo methods.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Abstract

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.     

Parameter Estimation for the Discrete Stable Family

The discrete stable family constitutes an interesting two-parameter model of distributions on the non-negative integers with a Paretian tail. The practical use of the discrete stable distribution is inhibited by the lack of an explicit expression for its probability function. Moreover, the distribution does not possess moments of any order. Therefore, the usual tools—such as the maximum-likelihood method or even the moment method—are not feasible for parameter estimation. However, the probability generating function of the discrete stable distribution is available in a simple form. Hence, we initially explore the application of some existing estimation procedures based on the empirical probability generating function. Subsequently, we propose a new estimation method by minimizing a suitable weighted L ²-distance between the empirical and the theoretical probability generating functions. In addition, we provide a goodness-of-fit statistic based on the same distance.     

Maximum term of a particular autoregressivesequence with discrete margins

In this paper we consider a stationary sequence of discrete random variables with marginal distribution H(x), obtained by a simple transformation from the max-AR(1) sequence considered by Alpuim (1989). Because discrete distributions impose severe restrictions on the convergence of the normalized maxima to an extreme value distribution, it is seen that in this particular case, whenever H(x) belongs to the domain of attraction of any max-stable distribution, the sequence possesses an extremal index 0 = 0. Nevertheless, it, is possible to obtain a nondegenerate limiting distribution for the linearized maxima by choosing other sets of normalizing constants. Whenever H(x) does not belong to the domain of attraction of any max-stable distribution, but, satisfies adequate conditions, the maxima nearly possess an asymptotic stability with the presence of an extremal index 0 <θ<1.

Motivated by the behaviour of these sequences we obtained a more general result extending the results of Anderson (1970) and Me (Jon nick and Park (1992) over the mixing conditionsD ^(k)(u_n), defined by Chermck et al (1991).

Several examples, obtained after simulation, are presented in order to illustrate the different situations that may occur.     

A generalized skew two-piece skew-normal distribution

In this paper, we discuss a general class of skew two-piece skew-normal distributions, denoted by GSTPSN(λ₁, λ₂, ρ). We derive its moment generating function and discuss some simple and interesting properties of this distribution. We then discuss the modes of these distributions and present a useful representation theorem as well. Next, we focus on a different generalization of the two-piece skew-normal distribution which is a symmetric family of distributions and discuss some of its properties. Finally, three well-known examples are used to illustrate the practical usefulness of this family of distributions.