For a wide class of discrete distributions, we derive a representation of the inverse (negative) moments through the Stirling numbers of the first kind and inverse factorial moments. We specialize the results for the Poisson, binomial, hypergeometric and negative binomial distributions.

Covariance structure analysis of nonnormal data is important because in practice all data are nonnormal. When applying covariance structure analysis to nonnormal data, it is generally assumed that the asymptotic covariance matrix Γ for the nonredundant terms in the sample covariance matrix S is...

The assumption of log-concavity is a flexible and appealing non-parametric shape constraint in distribution modelling. In this work, we study the log-concave maximum likelihood estimator of a probability mass function. We show that the maximum likelihood estimator is strongly consistent and we...

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{W_m\}{_{m\ge 1}}$$</EquationSource> </InlineEquation> be the sequence of weak records from a discrete parent random variable, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation>, supported on the non-negative integers. We obtain a new characterization of geometric distributions based on an additive property of weak records: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> </InlineEquation> follows a geometric...</equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation>

A new general method for generating discrete random variables is presented. The method is based on reducing the problem of generating a discrete random variable with an extremely large range to that of generating a random variable with a small range consisting of a few possible values...

It is often convenient to assume that X and X|Y are in the same exponential family. By considering X as the “parameter” and Y as the “data”, the problem becomes determining which exponential families X|Y have conjugate priors. We develop a necessary condition for conjugacy....

We extend the theory of penalty functions to stochastic programming problems with nonlinear inequality constraints dependent on a random vector with known distribution. We show that the problems with penalty objective, penalty constraints and chance constraints are asymptotically equivalent...

We extend the theory of penalty functions to stochastic programming problems with nonlinear inequality constraints dependent on a random vector with known distribution. We show that the problems with penalty objective, penalty constraints and chance constraints are asymptotically equivalent...

In this paper, we study several tests for the equality of two unknown distributions. Two are based on empirical distribution functions, three others on nonparametric probability density estimates, and the last ones on differences between sample moments. We suggest controlling the size of such...

In this paper, we study several tests for the equality of two unknown distributions. Two are based on empirical distribution functions, three others on nonparametric probability density estimates, and the last ones on differences between sample moments. We suggest controlling the size of such...