Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

BAYESIAN ANALYSIS FOR THE SUPERPOSITION OF TWO DEPENDENT NONHOMOGENEOUS POISSON PROCESSES

ABSTRACT

A Bayesian analysis for the superposition of two dependent nonhomogenous Poisson processes is studied by means of a bivariate Poisson distribution. This particular distribution presents a new likelihood function which takes into account the correlation between the two nonhomogenous Poisson processes. A numerical example using Markov Chain Monte Carlo method with data augmentation is considered.     

An extended Poisson distribution

ABSTRACT

The Poisson distribution is extended over the set of all integers. The motivation comes from the many reflected versions of the gamma distribution, the continuous analog of the Poisson distribution, defined over the entire real line. Various mathematical properties of the extended Poisson distribution are derived. Estimation procedures by the methods of moments and maximum likelihood are also derived with their performance assessed by simulation. Finally, a real data application is illustrated.     

Inferences on the difference between future observations for comparing two treatments

The comparison of two treatments with normally distributed data is considered. Inferences are considered based upon the difference between single potential future observations from each of the two treatments, which provides a useful and easily interpretable assessment of the difference between the two treatments. These methodologies combine information from a standard confidence interval analysis of the difference between the two treatment means, with information available from standard prediction intervals of future observations. Win-probabilities, which are the probabilities that a future observation from one treatment will be superior to a future observation from the other treatment, are a special case of these methodologies. The theoretical derivation of these methodologies is based upon inferences about the non-centrality parameter of a non-central t-distribution. Equal and unequal variance situations are addressed, and extensions to groups of future observations from the two treatments are also considered. Some examples and discussions of the methodologies are presented.     

Inference about parameters in Binomial-Poisson distribution with additional incomplete data

ABSTRACT

This article considers the distribution of Binomial-Poisson random vector which has two components and includes two parameters: one is the rate of a Poisson distribution, the other is the proportion in a Binomial distribution. The inference about the two parameters is usually made based on only paired observations. However, the number of paired observations is, in general, not large enough because of either technical difficulty or budget limitation, and so one can not make efficient inferences with only paired data. Instead, it is often much easier and not too costly to have incomplete observation on only one component independently. In this article we will combine both the paired complete data and unpaired incomplete data for estimating the two parameters. The performances of various estimators are compared both analytically and numerically. It is observed that fully using the unpaired incomplete data can always improve the inference, and the improvement is very significant in the case when there are only a few paired complete observations.     

Bias reduction for the maximum likelihood estimator of the doubly-truncated Poisson distribution

ABSTRACT

We derive an analytic expression for the bias of the maximum likelihood estimator of the parameter in a doubly-truncated Poisson distribution, which proves highly effective as a means of bias correction. For smaller sample sizes, our method outperforms the alternative of bias correction via the parametric bootstrap. Bias is of little concern in the positive Poisson distribution, the most common form of truncation in the applied literature. Bias appears to be the most severe in the doubly-truncated Poisson distribution, when the mean of the distribution is close to the right (upper) truncation.     

A Bayesian inference of P(λ1<λ2) for two Poisson parameters

The statistical inference drawn from the difference between two independent Poisson parameters is often discussed in the medical literature. However, such discussions are usually based on the frequentist viewpoint rather than the Bayesian viewpoint. Here, we propose an index θ=P(λ_{1, post}<λ_{2, post}), where λ_{1, post} and λ_{2, post} denote Poisson parameters following posterior density. We provide an exact and an approximate expression for calculating θ using the conjugate gamma prior and compare the probabilities obtained using the approximate and the exact expressions. Moreover, we also show a relation between θ and the p-value. We also highlight the significance of θ by applying it to the result of actual clinical trials. Our findings suggest that θ may provide useful information in a clinical trial.     

The Poisson-X family of distributions

ABSTRACT

Recently, Risti? and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.     

Compounded inverse Weibull distributions: Properties,inference and applications

ABSTRACT

In this paper two probability distributions are analyzed which are formed by compounding inverse Weibull with zero-truncated Poisson and geometric distributions. The distributions can be used to model lifetime of series system where the lifetimes follow inverse Weibull distribution and the subgroup size being random follows either geometric or zero-truncated Poisson distribution. Some of the important statistical and reliability properties of each of the distributions are derived. The distributions are found to exhibit both monotone and non-monotone failure rates. The parameters of the distributions are estimated using the expectation-maximization algorithm and the method of minimum distance estimation. The potentials of the distributions are explored through three real life data sets and are compared with similar compounded distributions, viz. Weibull-geometric, Weibull-Poisson, exponential-geometric and exponential-Poisson distributions.     

On Stein's method, smoothing estimates in total variation distance and mixture distributions

In many settings it is useful to have bounds on the total variation distance between some random variable Z and its shifted version Z+1. For example, such quantities are often needed when applying Stein's method for probability approximation. This note considers one way in which such bounds can be derived, in cases where Z is either the equilibrium distribution of some birth-death process or the mixture of such a distribution. Applications of these bounds are given to translated Poisson and compound Poisson approximations for Poisson mixtures and the Pólya distribution.     

Analysis of overdispersed paired count data

This article is about the statistical analysis of overdispersed paired count data for comparing two treatments. The data consist of the number of events obtained in a stratum during the fixed observation period. Three types of model are discussed: the Poisson, a mixed, and a semiparametric model. Overdispersion is represented in the last two models but not in the Poisson model. Of particular interests are to examine whether there is any loss of efficiency in using the estimate of the treatment effect obtained under other two models if the mixed model is true, and also whether overdispersion leads to a larger variance of the estimate than that expected from the Poisson model. It is shown that all three models provide the same estimate of the treatment effect (i.e., there is no loss of efficiency) and that the variance of the estimate of the treatment effect obtained under the Poisson model is the same as that based on the mixed model. However, the semiparametric model provides the variance of the estimate larger than those obtained under the other two models.     

Multi-Center Clinical Trials with Random Enrollment: Theoretical Approximations

ABSTRACT

We consider the problem of analyzing multi-center clinical trials when the number of patients at each center and on each treatment arm is random and follows the Poisson distribution. Theoretical approximations are made for the first two moments of the mean square errors (MSE's) for three different estimators of treatment effect difference that are commonly used in multi-center clinical trials. To construct these approximations, approximations are needed for the harmonic mean and negative moments of the Poisson distribution. This is achieved through the use of recurrence relations. The accuracy of the approximations for the moments of the MSE's were then validated through comparing the theoretical values to those obtained from a simulation study under two different enrollment environments.     

Analysis of over-dispersed count data with extra zeros using the Poisson log-skew-normal distribution

ABSTRACT

Mixed Poisson distributions are widely used in various applications of count data mainly when extra variation is present. This paper introduces an extension in terms of a mixed strategy to jointly deal with extra-Poisson variation and zero-inflated counts. In particular, we propose the Poisson log-skew-normal distribution which utilizes the log-skew-normal as a mixing prior and present its main properties. This is directly done through additional hierarchy level to the lognormal prior and includes the Poisson lognormal distribution as its special case. Two numerical methods are developed for the evaluation of associated likelihoods based on the Gauss–Hermite quadrature and the Lambert's W function. By conducting simulation studies, we show that the proposed distribution performs better than several commonly used distributions that allow for over-dispersion or zero inflation. The usefulness of the proposed distribution in empirical work is highlighted by the analysis of a real data set taken from health economics contexts.     

A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.     

A note on LeCam’s bound for the distance between the Poisson binomial and the Poisson distribution

n possibly different success probabilities p ₁, p ₂, ..., p _n is frequently approximated by a Poisson distribution with parameter λ = p ₁ + p ₂ + ... + p _n . LeCam's bound p ² ₁ + p ² ₂ + ... + p _n ² for the total variation distance between both distributions is particularly useful provided the success probabilities are small. The paper presents an improved version of LeCam's bound if a generalized d-dimensional Poisson binomial distribution is to be approximated by a compound Poisson distribution. Received: May 10, 2000; revised version: January 15, 2001     

Win-probabilities for comparing two binary outcomes

This article considers the problem of choosing between two treatments that have binary outcomes with unknown success probabilities p₁ and p₂. The choice is based upon the information provided by two observations X₁ ～ B(n₁, p₁) and X₂ ～ B(n₂, p₂) from independent binomial distributions. Standard approaches to this problem utilize basic statistical inference methodologies such as hypothesis tests and confidence intervals for the difference p₁ ? p₂ of the success probabilities. However, in this article the analysis of win-probabilities is considered. If X*₁ represents a potential future observation from Treatment 1 while X*₂ represents a potential future observation from Treatment 2, win-probabilities are defined in terms of the comparisons of X*₁ and X*₂. These win-probabilities provide a direct assessment of the relative advantages and disadvantages of choosing either treatment for one future application, and their interpretation can be combined with other factors such as costs, side-effects, and the availabilities of the two treatments. In this article, it is shown how confidence intervals for the win-probabilities can be constructed, and examples of their use are provided. Computer code for the implementation of this new methodology is available from the authors.     

Comparison of Poisson Confidence Intervals

Abstract

The standard method of obtaining a two-sided confidence interval for the Poisson mean produces an interval which is exact but can be shortened without violating the minimum coverage requirement. We classify the intervals proposed as alternatives to the standard method interval. We carry out the classification using two desirable properties of two-sided confidence intervals.     

The Least Upper Bound on the Poisson-Negative Binomial Relative Error

In this article, basic mathematical computations are used to determine the least upper bound on the relative error between the negative binomial cumulative distribution function with parameters n and p and the Poisson cumulative distribution function with mean λ =nq = n(1 ? p). Following this bound, it is indicated that the negative binomial cumulative distribution function can be properly approximated by the Poisson cumulative distribution function whenever q is sufficiently small. Five numerical examples are presented to illustrate the obtained result.     

Analysis of survival data by a Weibull-generalized Poisson distribution

In life-testing and survival analysis, sometimes the components are arranged in series or parallel system and the number of components is initially unknown. Thus, the number of components, say Z, is considered as random with an appropriate probability mass function. In this paper, we model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model: increasing, decreasing, bathtub and upside bathtub failure rates. Two examples are provided and the maximum-likelihood estimation of the parameters is studied. Rao's score test is developed to compare the results with the exponential Poisson model studied by Kus [17] and the exponential-generalized Poisson distribution with baseline distribution as exponential and the distribution of Z as generalized Poisson. Simulation studies are carried out to examine the performance of the estimates.