Mixture of the Riesz Distribution with Respect to a Multivariate Poisson

The aim of this article is to study a statistical model obtained by the mixture of the Riesz probability distribution on symmetric matrices with respect to a multivariate Poisson distribution. We show that this distribution is related to the modified Bessel function of the first kind. We then determine the domain of the means and the variance function of the generated natural exponential family.     

Use of Quasi-Ranges in the Estimation of Scale Parameter of a Symmetric Distribution

In this article, we consider the problem of best linear unbiased estimation and best linear invariant estimation of the scale parameter of a symmetric distribution using quasi-ranges is considered. We also prove a sufficient condition for the non negativity of the scale estimator obtained by the above method. Further, we obtain necessary and sufficient conditions for the derived estimators to be constant multiple of the sample range.     

Sufficient Reduction in Multivariate Surveillance

The relation between change points in multivariate surveillance is important but seldom considered. The sufficiency principle is here used to clarify the structure of some problems, to find efficient methods, and to determine appropriate evaluation metrics. We study processes where the changes occur simultaneously or with known time lags. The surveillance of spatial data is one example where known time lags can be of interest. A general version of a theorem for the sufficient reduction of processes that change with known time lags is given. A simulation study illustrates the benefits or the methods based on the sufficient statistics.     

The Poisson-conjugate Lindley mixture distribution

ABSTRACT

A new discrete distribution that depends on two parameters is introduced in this article. From this new distribution the geometric distribution is obtained as a special case. After analyzing some of its properties such as moments and unimodality, recurrences for the probability mass function and differential equations for its probability generating function are derived. In addition to this, parameters are estimated by maximum likelihood estimation numerically maximizing the log-likelihood function. Expected frequencies are calculated for different sets of data to prove the versatility of this discrete model.     

On the optimality of the sample variance

The present paper explores the structure of linear exponential families for which the sample variance is a uniformly minimum variance unbiased estimator.     

Maximum likelihood ridge displays

I illustrate likelihood methods for estimating the consequences of shrinkage along any ridge path as well as methods for picking a two-hyperparameter path of optimal curvature and the optimal point on that path. In addition to my published "classical" methods, I also illustrate both the empirical Bayes and the random coefficient maximum likelihood approaches. Traces of risks for known parameters and losses for simulated responses are followed by traces of estimates that can reveal the same general information.     

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.

     

Second order efficiency of the ban-estimators for the multinomial family

Rao (1961, 1963) introduced a measure of second order efficiency (s.o.e.) of a best asymptotically normal (BAN) estimator and obtained the s.o.e's of some well known estimators of the parameter of the multinomial family. Koorts (1985) dealt with a calss of BAN estimators and derived the s.o.e, of the estimator belonging to this class. In this paper we derive a general expressiion for the s.o.e. of a BAN estiimator based on its estimating equation.     

Analysing exponential family based messy data

Recently exponential family based random effects models have received considerable attention. These models usually arise from an unobservable random process added to the independent exponential family models. An unobservable correlated process, however, would cause correlations among the exponential family based data. This paper, first, develops an asymptotically optimal test for testing the appropriateness of a fixed effects model for the exponential family based independent data versus a random effects model for the exponential family based independent or correlated data. The paper, then, provides a general framework on regression analysis for the exponential family based data generated under the random effects models.     

Convergence rates for empirical bayes estimators of parameters in multi-parameter exponential families

This paper obtains the convergence rates of the empirical Bayes estimators of parameters in the multi-parameter exponential families. The rates can approximate to 0(n=¹) arbitrarily. The paper presents the multivariate orthogonal polynomials which are continuous on the total space R^p.