Descriptive statistics for multivariate distributions

The purpose of this paper is to study the concepts location, scatter, skewness and kurtosis of multivariate distributions. Measures of these properties are introduced which include some new generalizations of well-known univariate statistics. Previous work is briefly reviewed.     

The generalized beta- and F-distributions in statistical modelling

The univariate generalized beta- and generalized F-distributions are frequently in recent statistical modellings and applications. They have richer properties than the standard beta- and Snedecor F-distributions and provide more flexibility than these distributions, of which they are natural extensions. Their connection with the Gauss hypergeometric function and Lauricella functions leads to further generalizations and important properties. This article gives a unified and up-to-date treatment of these two generalized distributions using only simple arguments. Proofs are given for some original results and a complete reference to their source is provided for established ones. The important problem of parameter estimation is also studied.     

Infinitely divisible distributions over locally compact non-archimedean fields

This paper is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite, locally compact fields of zero and positive characteristics with nontrivial non-Archimedean norms. Infinitely divisible distributions are studied. Theorems about their characteristic functionals are proved. Particular cases are demonstrated as applications to non-Archimedean analogs of Gaussian and Poisson processes and their generalizations.     

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.     

Simulation of certain multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently. For further progress with these distributions simulation methods are an important part. We describe several methods of simulating GPDs, beginning with an efficient method for the logistic GPD. The algorithm is based on the Shi transformation, which was already used for the simulation of multivariate extreme value distributions (EVDs) of logistic type. In the sequel another algorithm is presented simulating a broader class of GPDs. Due to its numerical complexity it is only practicably applicable in low dimensions. A method is given to generate unconditional GPD random vectors from conditionally GPD distributed random vectors. A short application of the simulation methods in the analysis of a real hydrological data set concludes the article. The simulation algorithms are available on the author’s home page .      

The limiting distributions of some subclasses of the generalized non-centralt-distribution

A general theorem on the limiting distribution of the generalizedt-distribution is obtained, many applications of this theorem to some subclasses of elliptically contoured distributions including multivariate normal and multivariatet distributions are discussed. Further, their limiting distributions by density function are derived.This project is supported by the National Natural Science Foundation of China and by Grant DA01070 from U.S. Public Health Service.     

Some bivariate gamma distributions

We introduce two new bivariate gamma distributions based on a characterizing property involving products of gamma and beta random variables. We derive various representations for their joint densities, product moments, conditional densities and conditional moments. Some of these representations involve special functions such as the complementary incomplete gamma and Whittaker functions. We also discuss ways to construct multivariate generalizations.     

Hierarchical orbital decompositions and extended decomposable distributions

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki [Star-shaped distributions and their generalizations, J. Statist. Plann. Inference, 2006, available at 〈http://arxiv.org/abs/math.ST/0605600〉, to appear] generalized the elliptically contoured distributions to star-shaped distributions, for which the contours are allowed to be arbitrary proportional star-shaped sets. This was achieved by considering the so-called orbital decomposition of the sample space in the general framework of group invariance. In the present paper, we extend their results by conducting the orbital decompositions in steps and obtaining a further, hierarchical decomposition of the sample space. This allows us to construct probability models and distributions with further independence structures. The general results are applied to the star-shaped distributions with a certain symmetric structure, the distributions related to the two-sample Wishart problem and the distributions of preference rankings.     

On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula for the mode of the generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.     

Wrapped weighted exponential distributions

We have developed a new class of circular distributions named wrapped weighted exponential distributions. The estimation of unknown parameters along with some characteristics of these distributions is also investigated. Some theorems that relate the distribution to some other circular distributions are established and we clarify their modeling potential using a classical data set on movements of sea stars.     

Approximation with generalized hyperexponential distributions: Weak convergence results

Generalized hyperexponential (GH) distributions are linear combinations of exponential CDFs with mixing parameters (positive and negative) that sum to unity. The denseness of the class GH with respect to the class of all CDFs defined on [0, ) is established by showing that a GH distribution can be found that is as close to a given CDF as desired, with respect to a suitably defined metric. The metric induces the usual topology of weak convergence so that, equivalently, there exists a sequence of GH CDFs that converges weakly to a given CDF. This result is established by using a similar result for weak convergence of Erlang mixtures. Various set inclusion relations are also obtained relating the GH distributions to other commonly used classes of approximating distributions, including generalized Erlang (GE), mixed generalized Erlang (MGE), those with reciprocal polynomial Laplace transforms (K _n ), those with rational Laplace transforms (R _n ), and phase-type (PH) distributions. A brief survey of the history and use of approximating distributions in queueing theory is also included.This research was partially supported by the Office of Naval Research under Contract No. N00014-86-K0029. Much of this work is taken from the first-named author's doctoral dissertation, accepted by the faculty at the University of Virginia.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Structure theorems for associated homogeneous distributions based on the line

Associated homogeneous distributions (AHDs) with support in the line R are the distributional generalizations of one‐dimensional power‐log functions. In this paper, we derive a number of practical structure theorems for AHDs based on R and being complex analytic with respect to their degree of homogeneity in some region of the complex plane. Each theorem gives a representation that is designed to have a distinct advantage for calculating either convolution products, multiplication products, generalized derivatives and primitives, Fourier transforms or Hilbert transforms of AHDs. Copyright © 2008 John Wiley & Sons, Ltd.     

The skew elliptical distributions and their quadratic forms

In this paper, a family of the skew elliptical distributions is defined and investigated. Some basic properties, such as stochastic representation, marginal and conditional distributions, distribution under linear transformations, moments and moment generating function are derived. The joint distribution of several quadratic forms is obtained. An example is given to show that the distributions of some statistics as the functions of the quadratic forms can be derived for various applications.     

A space of generalized distributions

In this paper we use a duality method to introduce a new space of generalized distributions. This method is exactly the same introduced by Schwartz for the distribution theory. Our space of generalized distributions contains all the Schwartz distributions and all the multipole series of physicists and is, in a certain sense, the smallest space containing all these series. To The Memory of Laurent Schwartz     

The generalized hypergeometric family of distributions

Summary This is an expository summary of the authors' report on classification of the generalized hypergeometric (GHg for short) family of distributions (Sibuya and Shimizu (1981),Keio Science and Technology Report, to appear). Emphasis is laid on the definition of the distributions based on some conventional rules, and on the complete classification of the multivariate GHg distributions, whose types are found to be rather limited in spite of their quite general definition. Previous classifications and namings are summarized and compared with the new one.     

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.     

Flexible bivariate beta distributions

Bivariate beta distributions which can be used to model data sets exhibiting positive or negative correlation are introduced. Properties of these bivariate beta distributions and their applications in Bayesian analysis are discussed. Three methods for parameter estimation are presented. The performance of these estimators is evaluated based on Monte Carlo simulations. Examples are provided to illustrate how additional parameters can be introduced to gain even more modeling flexibility. A possible extension of the proposed bivariate beta model and a multivariate generalization are also discussed.     

Sequences of elliptical distributions and mixtures of normal distributions

Two conditions are shown under which elliptical distributions are scale mixtures of normal distributions with respect to probability distributions. The issue of finding the mixing distribution function is also considered. As a unified theoretical framework, it is also shown that any scale mixture of normal distributions is always a term of a sequence of elliptical distributions, increasing in dimension, and that all the terms of this sequence are also scale mixtures of normal distributions sharing the same mixing distribution function. Some examples are shown as applications of these concepts, showing the way of finding the mixing distribution function.     

On a generalized mixture of standard normal and skew normal distributions

Here we propose a new class of distributions as a generalized mixture of standard normal and skew normal distributions (GMNSND) and study some of its properties by deriving its characteristic function, mean, variance, coefficient of skewness etc. Further, certain reliability aspects of GMNSND are studied and a location scale extension of GMNSND is considered. The estimation of the parameters of this extended GMNSND by the method of maximum likelihood is discussed.