Asymptotic distributions of weighted divergence between discrete distributions

A divergence measure between discrete probability distributions introduced by Csiszar (1967) generalizes the Kullback-Leibler information and several other information measures considered in the literature. We introduce a weighted divergence which generalizes the weighted Kullback-Leibler information considered by Taneja (1985). The weighted divergence between an empirical distribution and a fixed distribution and the weighted divergence between two independent empirical distributions are here investigated for large simple random samples, and the asymptotic distributions are shown to be either normal or equal to the distribution of a linear combination of independent X²-variables     

Simulating from polynomial-normal distributions

The family of polynomial-normal distributions includes a number of widely used distributions, such as the Gram–Charlier and Edgeworth distributions. In this note, we present three simple algorithms, (i) CDF Inversion, (ii) Acceptance–Rejection, (iii) and Ratio–of–Uniforms, for simulating variates from a polynomial-normal distribution. Details on the efficiency of the Acceptance–Rejection and the Ratio–of–Uniforms algorithms and a comparison across the various implementations are provided.     

Riesz inverse Gaussian distributions

The notion of generalized power of a positive definite symmetric matrix and a related notion of generalized Bessel function are used to introduce an extension of the class of matrix generalized inverse Gaussian distributions. The new distributions are shown to arise as conditional distributions of Peirce components of Riesz random matrices. Things are explained in the modern framework of symmetric cones and simple Euclidean Jordan algebra.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

Asymmetric tolerance distributions: an application of two-piece distributions

A technique is proposed which allows for the fitting of and testing for asymmetric tolerance distributions in quantal assays. The implementation of the technique in statistical packages is discussed and is illustrated by an application to a simple data set.     

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,     

Minimally informative distributions with given rank correlation for use in uncertainty analysis

Minimum information bivariate distributions with uniform marginals and a specified rank correlation are studied in this paper. These distributions play an important role in a particular way of modeling dependent random variables which has been used in the computer code UNICORN for carrying out uncertainty analyses. It is shown that these minimum information distributions have a particular form which makes simulation of conditional distributions very simple. Approximations to the continuous distributions are discussed and explicit formulae are determined. Finally a relation is discussed to DAD theorems, and a numerical algorithm is given (which has geometric rate of covergence) for determining the minimum information distributions.     

Back-to-back mixtures of discrete distributions

A method for expanding the choices for fits of discrete data is given. The method is very simple: a breakpoint is chosen for the data set on either side of which two separate discrete distributions are fit. Thus, the method is a mixture of two discrete distributions. The method is appealing in light of the ease with which the likelihood equations simplify. For illustrative purposes, the method is used on the data set that motivated its conception.     

Inverse Weibull power series distributions: properties and applications

Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

Conditional even point estimation for bivariate discrete distributions

This paper presents a simple procedure for estimating the parameters of bivariate discrete distributions. The procedure uses the marginal means and certain observed frequencies in one or more conditional distributions. The bivariate Poisson and Negative Binomial distributions are used as illustrative examples, Parameter estimators are derived and asymptotic efficiencies are examined for various parameter values.     

The effects on the distributions of sample canonical correlations under nonnormality

In this paper, we study the effects of nonnormality on the distributions of sample canonical correlations when the population canonical correlations are simple. In order to achieve the purpose, we derive asymptotic expansion formulas for the distributions of a function of the canonical correlations as well as the individual canonical correlations under nonnormal populations. We particularly discuss the distribution of sample canonical correlations under the class of elliptical population. These expansions are given by using a perturbation method. Simulation results are also given.     

Optimal collapsing of mixture distributions in robust recursive estimation

Several authors have discussed Kalman filtering procedures using a mixture of normals as a model for the distributions of the noise in the observation and/or the state space equations. Under this model, resulting posteriors involve a mixture of normal distributions, and a “collapsing method” must be found in order to keep the recursive procedure simple. We prove that the Kullback-Leibler distance between the mixture posterior and that of a single normal distribution is minimized when we choose the mean and variance of the single normal distribution to be the mean and variance of the mixture posterior. Hence, “collapsing by moments” is optimal in this sense. We then develop the resulting optimal algorithm for “Kalman filtering” for this situation, and illustrate its performance with an example.     

A family of bivariate binomial distributions generated by extreme bernoulli distributions

In this paper, bivariate binomial distributions generated by extreme bivariate Bernoulli distributions are obtained and studied. Representation of the bivariate binomial distribution generated by a convex combination of extreme bivariate Bernoulli distributions as a mixture of distributions in the class of bivariate binomial distribution generated by extreme bivariate Bernoulli distribution is obtained. A subfamily of bivariate binomial distributions exhibiting the property of positive and negative dependence is constructed. Some results on positive dependence notions as it relates to the bivariate binomial distribution generated by extreme bivariate Bernoulli distribution and a linear combination of such distributions are obtained.     

Polynomial structures in rank statistics distributions

This paper deals with the classical problem of how to evaluate the joint rank statistics distributions for two independent i.i.d. samples from a common continuous distribution. It is pointed that these distributions rely on an underlying polynomial structure of negative binomial type. That property is exploited to obtain, in a systematic and unified way, closed forms and simple recursions, some well established, for computing the joint tail and rectangular probabilities of interest.     

Multivariate geometric distributions

Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping roles for different counts, A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Goldberg (1934), Bates and Meyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppulori (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation is provided.     

Complex elliptically symmetric distributions

In this paper, the authors introduce a class of distributions known as complex elliptically symmetric distributions. The complex multivariate normal and complex multivariate t distributions are members of this class. Various properties of the complex elliptically symmetric distributions are studied. Finally, the robustness of certain test procedures are discussed when the assumption of complex multivariate normality is violated but the underlying distribution still belongs to the class of elliptically symmetric distributions.     

Truncated-exponential skew-symmetric distributions

The family of skew distributions introduced by Azzalini and extended by others has received widespread attention. However, it suffers from complicated inference procedures. In this paper, a new family of skew distributions that overcomes the difficulties is introduced. This new family belongs to the exponential family. Many properties of this family are studied, inference procedures developed and simulation studies performed to assess the procedures. Some particular cases of this family, evidence of its flexibility and a real data application are presented. At least 10 advantages of the new family over Azzalini's distributions are established.     

On predictive distributions and Bayesian networks

In this paper we are interested in discrete prediction problems for a decision-theoretic setting, where the task is to compute the predictive distribution for a finite set of possible alternatives. This question is first addressed in a general Bayesian framework, where we consider a set of probability distributions defined by some parametric model class. Given a prior distribution on the model parameters and a set of sample data, one possible approach for determining a predictive distribution is to fix the parameters to the instantiation with the maximum a posteriori probability. A more accurate predictive distribution can be obtained by computing the evidence (marginal likelihood), i.e., the integral over all the individual parameter instantiations. As an alternative to these two approaches, we demonstrate how to use Rissanen's new definition of stochastic complexity for determining predictive distributions, and show how the evidence predictive distribution with Jeffrey's prior approaches the new stochastic complexity predictive distribution in the limit with increasing amount of sample data. To compare the alternative approaches in practice, each of the predictive distributions discussed is instantiated in the Bayesian network model family case. In particular, to determine Jeffrey's prior for this model family, we show how to compute the (expected) Fisher information matrix for a fixed but arbitrary Bayesian network structure. In the empirical part of the paper the predictive distributions are compared by using the simple tree-structured Naive Bayes model, which is used in the experiments for computational reasons. The experimentation with several public domain classification datasets suggest that the evidence approach produces the most accurate predictions in the log-score sense. The evidence-based methods are also quite robust in the sense that they predict surprisingly well even when only a small fraction of the full training set is used.     

A counterexample to a claim concerning the convolution of multimodal distributions

P. Ghosh (1981) has claimed that the convolution of two symmetric multimodal distributions is symmetric and unimodal. A simple counterexample to this claim is constructed by considering the convolution f?f, where f is an appropriate mixture of two normal densities.