Perfect simulation of conditionally specified models

We discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as auto-gamma, auto-Poisson and autonegative binomial models, using Gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes.     

Admissibility of generalized bayes and pitman estimates in the non-regular family

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities Following Karlin s (1958) technique, we have established the ad-missibility of generalized Bayes estimates and Pitman estimates. Some examples are discussed.     

Generalized linear models with random effects in the two-parameter exponential family

In this paper, we develop a new class of double generalized linear models, introducing a random-effect component in the link function describing the linear predictor related to the precision parameter. This is a useful procedure to take into account extra variability and also to make the model more robust. The Bayesian paradigm is adopted to make inference in this class of models. Samples of the joint posterior distribution are drawn using standard Monte Carlo Markov Chain procedures. Finally, we illustrate this algorithm by considering simulated and real data sets.     

Asymptotic non-null distribution of a test of equality of exponential populations

In this paper further asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained when the alternatives are close to the hypothesis. These expansions are obtained for the first time in terms of beta distributions.     

Note on sums of a random number of random variables with exponential family distributions

With the help of the result that exponential-type families are determined by their mean value functions it is shown that stochastic independence of the random variables S_N and N-S_N characterizes the Poisson and Bernoulli distributions simultaneously.     

A unified approach to minimum variance unbiased estimation of a probability function belonging to an exponential family

The problem of minimum variance unbiased estimation of the probability density function of a random variable belonging to an exponential family is considered. The method of estimation proposed in this paper requires the solution of a certain integral equation. For many probability distributions the solution of this equation is given by a known result in integral transform theory.     

On the robustness of the extreme deviate test for a single multivariate outlier against heavy-tailed distributions

In this paper we assess the sensitivity of the multivariate extreme deviate test for a single multivariate outlier to non-normality in the form of heavy tails. We find that the empirical significance levels can be markedly affected by even modest departures from multivariate normality. The effects are particularly severe when the sample size is large relative to the dimension. Finally, by way of example we demonstrate that certain graphical techniques may prove useful in identifying the source of rejection for the multivariate extreme deviate test.     

Role of concomitants of order statistics in determining parent bivariate distributions

In this paper, we establish the role of concomitants of order statistics in the unique identification of the parent bivariate distribution. From the results developed, we have illustrated by examples the process of determination of the parent bivariate distribution using a marginal pdf and the pdf of either of the concomitant of largest or smallest order statistic on the other variable. An application of the results derived in modeling of a bivariate distribution for data sets drawn from a population as well is discussed.     

Series evaluation of Tweedie exponential dispersion model densities

Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to theTweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.     

Generalizations of a contamination model for continuous type random variables

In 1965, Stanley Warner (Warner, 1965) introduced a model for contaminating discrete type random variables. He presented this scheme as being potentially useful in survevs where sensitive in-formation is being gathered. Since that time much research has been conducted and many papers written on the development of these discrete type randomized response models. More recently, atten-tion has been focused on the application of randomized response type models for preservation of confidentiality in existing data files (Boruch 1971 and 1972, Ranney 1975, Felligi 1974, and Inge-marsson 1975). In 1974, Poole (Poole, 1974) introduced a randomized response model for a positive continuous type random variable which was basically a continuous variable analog of the discrete variable Warner model. In this paper the results of the 1974 paper are extended to a lt-dimensional continuous type random variable in k-dimensional Euclidean space.     

Characterization of a mixture of gamma distributions via conditional finite moments

A necessary and sufficient condition that a continuous, positive random variable follow a gamma distribution is given in terms of any one of its conditional finite moments and an expression involving its failure rate. The results are then used to develop a characterization for a mixture of two gamma distributions. The general results about characterization of a mixture of gamma distributions yield several special cases that have appeared separately in recent literature, including characterization of a single exponential distribution, characterization of a single gamma distribution (in terms of either first or second moments) and a sufficient condition for a mixture of two exponential distributions (in terms of first moments). The condition in this last result is shown to be necessary also. Numerous other cases are possible, using different choices for distribution parameters along with a selection of the mixing parameter, for either individual or mixtures of distributions. Various characterizations can be expressed using higher order moments, too.     

On the robustness of the power function of the one-sample test for the negative exponential distribution

The robustness of the power function of the standard one-sample parametric test for the mean of the negative exponential distribution is examined. The main form of departure from the exponential assumption is a mixture of negative exponential components although an alternative Gamma distribution is also examined. It is found that the test is sensitive to these departures although the effect of mixtures with short tails is less dramatic than those with long tails.     

Asymptotic non-null distribution of a test connected with exponential populations

In this paper asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained under local alternatives. These expansions are in terms of Chi-square distributions.     

A class of maximum-entropy multivariate distributions

This paper characterizes a class of multivariate distributions that includes the multinormal and is contained in the exponential family. The wide range of possible applications of these distributions is suggested by some of hte characteristics germane to them: First, they maximize Shannon's entropy among all distributions that have finite moments of given orders. As such, they constitute a class of distributions that includes the multinormal and some likely alternatives. Second, they can exhibit several modes, and, further-more, they do so with a relatively small number of parameters (compared to mixtures of multinormals). Third, they are the stationary distributions of certain diffusion processes. Fourth, they approximate, near the multinormal, the multivariate Pearson family. And fifth, the maximum likelihood estimators of their population moments are the sample moments. Two possible methods of estimating the distributions are studied in this paper: maximum likelihood estimation, and a fast procedure that can be used to find consistent estimators of the parameters via sample moments. A FORTTAN subroutine that implements the latter method is also provided.     

On extending bock's model of logistic regression in the analysis of categorical data

A general class of multiple logistic regression models is reviewed and an extension is proposed which leads to restricted maximum likelihood estimates of model parameters. Examples of thegeneral model are given, with an emphasis placed on the interpretation of the parameters in each case.     

The likelihood ratio under noncontiguous alternatives

The asymptotic distribution of the likelihood ratio under noncontiguous alternatives is shown to be normal for the exponential family of distributions. The rate of convergence of the parameters to the hypothetical value is specified where the asymptotic noncentral chi-square distribution no longer holds. It is only a little slower than $\O\left( {n^{ - \frac{1}{2}} } \right)$. The result provides compact power approximation formulae and is shown to work reasonably well even for moderate sample sizes.     

Centered parameterizations and dependence limitations in Markov random field models

Markov random field models incorporate terms representing local statistical dependence among variables in a discrete-index random field. Traditional parameterizations for models based on one-parameter exponential family conditional distributions contain components that would appear to reflect large-scale and small-scale model behaviors, and it is natural to attempt to match these structures with large-scale and small-scale patterns in a set of data. Traditional manners of parameterizing Markov random field models do not allow such correspondence, however. We propose an alternative centered parameterization that, while not leading to different models, allows a correspondence between model structures and data structures to be successfully accomplished. The ability to make these connections is important when incorporating covariate information into a model or if a sequence of models is fit over time to investigate and interpret possible changes in data structure. We demonstrate the improved interpretation that results from use of centered parameterizations. Centered parameterizations also lend themselves to computation of an interpretable decomposition of mean squared error, and this is demonstrated both analytically and through a simulated example. A breakdown in model behavior occurs even with centered parameterizations if dependence parameters in Markov random field models are allowed to become too large. This phenomenon is discussed and illustrated using an auto-logistic model.     

A Characterization of the Normal Family of Distributions Based on the Bias of the Maximum Likelihood Estimator

This paper characterizes the family of Normal distributions within the class of exponential families of distributions, via the structure of the bias of the maximum likelihood estimator Θ _n of the canonical parameter Θ . More specifically, when E _θ ( Θ n ) – Θ = (1/ n ) Q ( Θ ) + o (1/ n ), the equality Q ( Θ ) = 0 proves to be a property of the Normal distribution only. The same conclusion is obtained for the one-dimensional case bt assuming that Q ( Θ ) is a polynomial of Θ .