On the admissibility of ordered poisson parameter under the entropy loss function

Let X ₁, X ₂be two independent Poisson random variables with means θ ₁and θ ₂respectively. Assume 0 ≤ θ ₁ ≤ θ ₂ ≤ ∞. The problem is estimation of the ordered parameters which has received considerable attention in statistical literature during the last two decades, In this paper the main portion of the study is devoted to the problem of estimating the smallest of the two ordered Poisson means, when it is known which population corresponds to each mean under the entropy loss function. An extension of the problem to the k-ordered Poisson means is also discussed.     

The truncated generalized poisson distribution and its estimation

A discrete probability model always gets truncated during the sampling process and the point of truncation depends upon the sample size. Also, the generalized Poisson distribution cannot be used with full justification when the second parameter is negative. To avoid these problems a truncated generalized Poisson distribution is defined and studied. Estimation of its parameters by moments method, maximum likelihood method and a mixed method are considered. Some examples are given to illustrate the effect on the parameters’ estimates when a non-truncated GPD is used instead of a truncated GPD.     

On characterizing the normal and poisson distributions

Multivariate normal, correlated multivariate Poisson and multiple Poisson distributions are characterized, in the class of exponential-type distributions, by the properties of the linear combinations of the variables, the properties of their cumulants and the recurance relation between the cumulants.     

On comparing two poisson intensity functions

Given realizations of two possion processes with unknown intensities A(·) and F(·) observed over the interval (t₁,t₂), we suppose that it is desired to distinution between H₀ Ξ(·)/λ(·) is constant on (t₁,t₂) versus H₊:Ξ(·)/λ(·) increases on (t₁,t₂). We propose a decision rule which uses the percentage points of the Mann-Whitney U-distribution. We show that the decision rule is unbiased and that the set of alternatives in H₊ can be weakly ordered, specifically: if Ξ(·)/λ(·), β(·)/λ(·) and Ξ(·)/β(·) are increasing on (t₁, t₂) then P{H₀ is rejected |Ξ(·)}≧P{H₀ is rejected|B(·)}≧P{H₀ is rejected|H₀}.     

Estimation of poisson means under weighted squared error loss

In estimating the means of several independent Poisson distributions, we show that the maximum likelihood estimator is inadmissible when general weighted squared error loss is the criterion. Using this result, we extend the known results on estimation of several Poisson means (Peng 1975, Hudson 1978) to the case where possibly more than one observation is taken from each Poisson distribution and the samples are not necessarily of the same size.     

Error rates for poisson process discrimination

This research addresses the question of how one's ability to discriminate between two Poisson process models is affected by the relative behavior of the respective intensity functions involved. One sampling strategy studied involves observation of the process up to the k-th occurrence time, t_k. Another is to observe the process up to a fixed time T. The error probabilities for these two approaches are analyzed as k and T, respectively, tend to infinity.     

On some models leading to the generalized poisson distribution

A new generalization of the Poisson distribution was given by Consul and Jain (1970, 73). Since then more than twenty papers, written by various researchers, have appeared on this model under the titles of Generalized Poisson Distribution (GPD), Lagrangian Poisson distribution or modified power series distribution. Here the author provides two physical models, based on differential-difference equations, which lead to the GPD. A number of axioms are given for a steady state point process which produce the generalized Poisson process. Also, the GPD is derived as the limiting distribution of the two quasi-binomial distributions based on urn models.     

Semiparametric principal component poisson regression on clustered data

In modeling count data with multivariate predictors, we often encounter problems with clustering of observations and interdependency of predictors. We propose to use principal components of predictors to mitigate the multicollinearity problem and to abate information losses due to dimension reduction, a semiparametric link between the count dependent variable and the principal components is postulated. Clustering of observations is accounted into the model as a random component and the model is estimated via the backfitting algorithm. Simulation study illustrates the advantages of the proposed model over standard poisson regression in a wide range of scenarios.     

Analysis of variance with poisson responses

In this paper we develop an analysis of variance procedure for two-way classifications in which the responses are counts having independent Poisson distributions. The distinction is made between this problem and observational studies analyzed via contingency tabie models. The relationship between our procedure and one-way analysis of variance for Poisson variates similar to Fisher’s variance test is explained.     

Censored Kullback-Leibler Information and Goodness-of-Fit Test with Type II Censored Data

The Kulback-Leibler information has been considered for establishing goodness-of-fit test statistics, which have been shown to perform very well (Arizono & Ohta, 1989; Ebrahimi et al., 1992, etc). In this paper, we propose censored Kullback-Leibler information to generalize the discussion of the Kullback-Leibler information to the censored case. Then we establish a goodness-of-fit test statistic based on the censored Kullback-Leibler information with the type 2 censored data, and compare the test statistics with some existing test statistics for the exponential and normal distributions.     

AN ITERATIVE METHOD FOR ESTIMATION OF PARAMETERS IN A CLUSTERED SAMPLING MODEL

This paper considers an iterative method for obtaining maximum likelihood estimates for a contingency table derived from a clustered sampling model. Comparisons are made with other methods proposed in the literature.     

Bivariate compound poisson distributions

This paper discusses four alternative methods of forming bivariate distributions with compound Poisson marginals. Basic properties of each bivariate version are given. A new bivariate negative binomial distribution, and four bivariate versions of the Sichel distribution, are defined and their properties given.     

Heine-euler extensions of the poisson distribution

The paper shows that the Heine and Euler distributions (Benkherouf and Bather, 1988) are members of a family of q-series anologues of the Poisson distribution, with similar probability mass functions, but different restrictions on their parameters, and different modes of genesis and properties. The relationships between the Heine, Euler, pseudo-Euler, Poisson and geometric distributions are explored. Illustrative data sets are discussed.     

On estimation of the scale matrix of the multivariate f distribution

Let F _p×phave a multivariate F distribution with a scale p×p matrix Δ and degrees of freedom k₁ and k₂ such that k_i - p - 1 > 0, i = 1,2. The estimation of Δ under entropy and squared error loss functions are considered. In both cases a new class of orthogonally invariant estimators are obtained which dominate the best unbiased estimator.     

Approximating the distribution of a renewal process using generalized poisson distributions

The exact distribution of a renewal counting process is not easy to compute and is rarely of closed form. In this article, we approximate the distribution of a renewal process using families of generalized Poisson distributions. We first compute approximations to the first several moments of the renewal process. In some cases, a closed form approximation is obtained. It is found that each family considered has its own strengths and weaknesses. Some new families of generalized Poisson distributions are recommended. Theorems are obtained determining when these variance to mean ratios are less than (or exceed) one without having to find the mean and variance. Some numerical comparisons are also made.     

Bayesian inference based on a jointly type-II censored sample from two exponential populations

In this paper, based on a jointly type-II censored sample from two exponential populations, the Bayesian inference for the two unknown parameters are developed with the use of squared-error, linear-exponential and general entropy loss functions. The problem of predicting the future failure times, both point and interval prediction, based on the observed joint type-II censored data, is also addressed from a Bayesian viewpoint. A Monte Carlo simulation study is conducted to compare the Bayesian estimators with the maximum likelihood estimator developed by Balakrishnan and Rasouli [Exact likelihood inference for two exponential populations under joint type-II censoring. Comput Stat Data Anal. 2008;52:2725–2738]. Finally, a numerical example is utilized for the purpose of illustration.