Generalized hypergeometric,digamma and trigamma distributions

Summary In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Pólya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions. The family of digamma distributions has the logarithmic series as a limit and the trigamma as another limit. The trigamma distributions are very close to the zeta (Zipf) distributions. Thus, our new distributions are useful as substitutes of the logarithmic series when the observed frequency data have such a long tail that cannot be fitted by the latter distributions. In the beginning sections we summarize properties of the Type B3 generalized hypergeometric distributions. It is emphasized that the distributions are obtained by compounding a Poisson distribution by “gamma product-ratio” distributions.     

An order-statistics-based method for constructing multivariate distributions with fixed marginals

A new system of multivariate distributions with fixed marginal distributions is introduced via the consideration of random variates that are randomly chosen pairs of order statistics of the marginal distributions. The distributions allow arbitrary positive or negative Pearson correlations between pairs of random variates and generalise the Farlie–Gumbel–Morgenstern distribution. It is shown that the copulas of these distributions are special cases of the Bernstein copula. Generation of random numbers from the distributions is described, and formulas for the Kendall and grade (Spearman) correlations are given. Procedures for data fitting are described and illustrated with examples.     

Semi-self-decomposable distributions on Z<Subscript>+</Subscript>

We present a notion of semi-self-decomposability for distributions with support in Z ₊. We show that discrete semi-self-decomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of discrete semi-self-decomposable distributions is shown to contain the discrete semistable distributions and the discrete geometric semistable distributions. We identify a proper subclass of semi-self-decomposable distributions that arise as weak limits of subsequences of binomially thinned sums of independent Z ₊-valued random variables. Multiple semi-self-decomposability on Z ₊ is also discussed.     

On modeling failure and repair times in stochastic models of manufacturing systems using generalized exponential distributions

Failures of machines have a significant effect on the behavior of manufacturing systems. As a result it is important to model this phenomenon. Many queueing models of manufacturing systems do incorporate the unreliability of the machines. Most models assume that the times to failure and the times to repair of each machine are exponentially distributed (or geometrically distributed in the case of discrete-time models). However, exponential distributions do not always accurately represent actual distributions encountered in real manufacturing systems. In this paper, we propose to model failure and repair time distributions bygeneralized exponential (GE) distributions (orgeneralized geometric distributions in the case of a discretetime model). The GE distribution can be used to approximate distributions with any coefficient of variation greater than one. The main contribution of the paper is to show that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential. Indeed, we show that failures and repair times represented by GE distributions can (under certain assumptions) be equivalently represented by exponential distributions.This work was performed while the author was visiting the Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.     

Sparse pseudorandom distributions

The existence of sparse pseudorandom distributions is proved. These are probability distributions concentrated in a very small set of strings, yet it is infeasible for any polynomial-time algorithm to distinguish between truly random coins and coins selected according to these distributions. It is shown that such distributions can be generated by (nonpolynomial) probabilistic algorithms, while probabilistic polynomial-time algorithms cannot even approximate all the pseudorandom distributions. Moreover, we show the existence of evasive pseudorandom distributions which are not only sparse, but also have the property that no polynomial-time algorithm may find an element in their support, except for a negligible probability. All these results are proved independently of any intractability assumption.     

Characteristic functions of scale mixtures of multivariate skew-normal distributions

We obtain the characteristic function of scale mixtures of skew-normal distributions both in the univariate and multivariate cases. The derivation uses the simple stochastic relationship between skew-normal distributions and scale mixtures of skew-normal distributions. In particular, we describe the characteristic function of skew-normal, skew-t, and other related distributions.     

Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals

The asymptotic optimal scaling of random walk Metropolis (RWM) algorithms with Gaussian proposal distributions is well understood for certain specific classes of target distributions. These asymptotic results easily extend to any light tailed proposal distribution with finite fourth moment. However, heavy tailed proposal distributions such as the Cauchy distribution are known to have a number of desirable properties, and in many situations are preferable to light tailed proposal distributions. Therefore we consider the question of scaling for Cauchy distributed proposals for a wide range of independent and identically distributed (iid) product densities. The results are somewhat surprising as to when and when not Cauchy distributed proposals are preferable to Gaussian proposal distributions. This provides motivation for finding proposal distributions which improve on both Gaussian and Cauchy proposals, both for finite dimensional target distributions and asymptotically as the dimension of the target density, d → ∞. Throughout we seek the scaling of the proposal distribution which maximizes the expected squared jumping distance (ESJD).     

Semi-selfdecomposable distributions and a new class of limit theorems

Self-decomposable distributions are given as limits of normalized sums of independent random variables. We define semi-selfdecomposable distributions as limits of subsequences of normalized sums. More generally, we introduce a way of making a new class of limiting distributions derived from a class of distributions by taking the limits through subsequences of normalized sums, and define the class of semi-selfdecomposable distributions and a decreasing sequence of subclasses of it. We give two kinds of necessary and sufficient conditions for distributions belonging to those classes, one is in terms of the decomposability of random variables and another is in terms of Lévy measures. Received: 1 May 1997 / Revised version: 5 February 1998     

The matrix-t distribution and its applications in predictive inference

The predictive distributions of the future responses and regression matrix under the multivariate elliptically contoured distributions are derived using structural approach. The predictive distributions are obtained as matrix-t which are identical to those obtained under matrix normal and matrix-t distributions. This gives inference robustness with respect to departures from the reference case of independent sampling from the matrix normal or dependent but uncorrelated sampling from matrix-t distributions. Some successful applications of matrix-t distribution in the field of spatial prediction have been addressed.     

A Subclass of Type G Selfdecomposable Distributions on ℝ d

A new class of type G selfdecomposable distributions on ℝ ^d is introduced and characterized in terms of stochastic integrals with respect to Lévy processes. This class is a strict subclass of the class of type G and selfdecomposable distributions, and in dimension one, it is strictly bigger than the class of variance mixtures of normal distributions by selfdecomposable distributions. The relation to several other known classes of infinitely divisible distributions is established. Research of J. Rosiński supported, in part, by a grant from the National Science Foundation.     

Selfdecomposability of moving average fractional Lévy processes

We study the relationships between the selfdecomposability of marginal distributions or finite dimensional distributions of moving average fractional Lévy processes and distributions of their driving Lévy processes.     

Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given.     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.     

Strictly operator-stable distributions

Strictly operator-stable distributions are defined and discussed. Characterization of strictly stable distributions with exponent 1 is generalized to strictly (α, Q)-stable distributions with α being an eigenvalue of Q.     

The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions

It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on ${\mathbb{R}^{d}}$ , which are the Jurek class, the Goldie– Steutel–Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type G distributions, are identical with the closure of the class of stable distributions. More general results are also given.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.     

Construction of a set of <Emphasis Type="Italic">p</Emphasis>-adic distributions

Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ?_p of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions).     

A bayes procedure for selecting the population with the largestpth quantile

Summary The Bayes method is seldom applied to nonparametric statistical problems, for the reason that it is hard to find mathematically tractable prior distributions on a set of probability measures. However, it is found that the Dirichlet process generates randomly a family of probability distributions which can be taken as a family of prior distributions for an application of the Bayes method to such problems. This paper presents a Bayesian analysis of a nonparametric problem of selecting a distribution with the largestpth quantile value, fromk≧2 given distributions. It is assumed a priori that the given distributions have been generated from a Dirichlet process. This work was supported by the U.S. Office of Naval Research under Contract No. 00014-75-C-0451.