The distribution of a moment estimator for a parameter of the generalized poisson distribution

The ratio of the sample variance to the sample mean estimates a simple function of the parameter which measures the departure of the Poisson-Poisson from the Poisson distribution. Moment series to order n?²⁴ are given for related estimators. In one case, exact integral formulations are given for the first two moments, enabling a comparison to be made between their asymptotic developments and a computer-oriented extended Taylor series (COETS) algorithm. The integral approach using generating functions is sketched out for the third and fourth moments. Levin's summation algorithm is used on the divergent series and comparative simulation assessments are given.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.     

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.     

NEYMAN SMOOTH TESTS FOR THE GENERALIZED PARETO DISTRIBUTION

ABSTRACT

The generalized Pareto distribution (GPD) is commonly used as extreme values's distribution. We present goodness of fit tests for the GPD based on Neyman's smooth tests statistics. The methods of maximum likelihood, moments and probability-weighted moments are used for estimating the GPD's parameters. Simulations are done to study the power of these tests.     

More Damned Lies and Statistics: How Numbers Confuse Public Issues and Chance: A Guide to Gambling,Love, the Stock Market, & Just About Everything Else

In this article we review two historical approximations to the Poisson and binomial cumulative distribution functions (CDFs); that is, the Wilson–Hilferty and Camp–Paulson approximations. Both of these approximations reduce to standard normal formulas that produce very accurate estimates of the Poisson and binomial CDFs, and are thus quite simple to implement. Additionally, in an upper-division undergraduate or master’s level probability and inference course, the derivation of these approximations presents a nice opportunity to introduce and study the distributional relationships between the gamma and Poisson CDFs, and the binomial, beta, and F CDFs. This article presents the basic theorems and lemmas needed to derive each approximation, along with some relevant examples that compare and contrast the precision of these approximations with their large-sample, limiting normal counterparts.     

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.     

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.     

Continuity corrected approximations for and ‘exact’ inference with Pearson's X2

The classical adjustments for the inadequacy of the asymptotic distribution of Pearson's X² statistic, when some cells are sparse or the cell expectations are small, use continuity corrections and exact moments; the recent approach is to use computer based ‘exact inference’. In this paper we observe that the original exact test due to Freeman and Halton (Biometrika 38 (1951), 141–149) and its computer implementation are theoretically unsound. Furthermore, the corrected algorithmic version for the exact p-value in StatXact is practically useful in very few cases, and the results of its present version which includes Monte Carlo estimates can be highly variable. We then derive asymptotic expansions for the moments of the null distribution of Pearson's X², introduce a new method of correcting for discreteness and finite range of Pearson's X² as an alternative to the classical continuity correction, and use them to construct new and improved approximations for the null distribution. We also offer diagnostic criteria applicable to the tables for selecting an appropriate approximation. The exact methods and the competing approximations are studied and compared using thirteen test cases from the literature. It is concluded that the accuracy of the appropriate approximation is comparable with the truly exact method whenever it is available. The use of approximations is therefore preferable if the truly exact computer intensive solutions are unavailable or infeasible.     

Optimal design of variable acceptance sampling plans for mixture distribution

ABSTRACT

The paper investigates the design of single and sequential variable acceptance sampling plans for a mixture distribution. Mixture distributions are seen in many practical problems such as life testing experiments of electronic components and clinical trials. The sampling plans for this kind of situations are not well addressed in the literature. We first propose a single sampling plan for a distribution which is a mixture of two exponential distributions. An optimization problem which minimizes the total cost of testing at given producer's and consumer's risks is solved to obtain the plan parameters. Two different sequential sampling plans are also defined and plan parameters are obtained by solving corresponding optimization problems. Finally, a case study, a simulation study and a sensitivity analysis are presented to illustrate our sampling plans.     

A generalization of the logistic linear'model

Consider the logistic linear model, with some explanatory variables overlooked. Those explanatory variables may be quantitative or qualitative. In either case, the resulting true response variable is not a binomial or a beta-binomial but a sum of binomials. Hence, standard computer packages for logistic regression can be inappropriate even if an overdispersion factor is incorporated. Therefore, a discrete exponential family assumption is considered to broaden the class of sampling models. Likelihood and Bayesian analyses are discussed. Bayesian computation techniques such as Laplacian approximations and Markov chain simulations are used to compute posterior densities and moments. Approximate conditional distributions are derived and are shown to be accurate. The Markov chain simulations are performed effectively to calculate posterior moments by using the approximate conditional distributions. The methodology is applied to Keeler's hardness of winter wheat data for checking binomial assumptions and to Matsumura's Accounting exams data for detailed likelihood and Bayesian analyses.     

Tests for detecting overdispersion in poisson models

Collings and Margolin(1985) developed a locally most powerful unbiased test for detecting negative binomial departures from a Poisson model, when the variance is a quadratic function of the mean. Kim and Park(1992) developed a locally most powerful unbiased test, when the variance is a linear function of the mean. It is found that a different mean-variance structure of a negative binomial derives a different locally optimal test statistic.

In this paper Collings and Margolin's and Kim and Park's results are unified and extended by developing a test for overdispersion in Poisson model against Katz family of distributions, Our setup has two extensions: First, Katz family of distributions is employed as an extension of the negative binomial distribution. Second, the mean-variance structure of the mixed Poisson model is given by σ² = μ+cμ^r for arbitrary but fixed r. We derive a local score test for testing H₀ : c = 0. Superiority of a new test is proved by the asymtotic relative efficiency as well as the simulation study.     

The exponential COM-Poisson distribution

The Conway-Maxwell Poisson (COMP) distribution as an extension of the Poisson distribution is a popular model for analyzing counting data. For the first time, we introduce a new three parameter distribution, so-called the exponential-Conway-Maxwell Poisson (ECOMP) distribution, that contains as sub-models the exponential-geometric and exponential-Poisson distributions proposed by Adamidis and Loukas (Stat Probab Lett 39:35?C42, 1998) and Ku? (Comput Stat Data Anal 51:4497?C4509, 2007), respectively. The new density function can be expressed as a mixture of exponential density functions. Expansions for moments, moment generating function and some statistical measures are provided. The density function of the order statistics can also be expressed as a mixture of exponential densities. We derive two formulae for the moments of order statistics. The elements of the observed information matrix are provided. Two applications illustrate the usefulness of the new distribution to analyze positive data.     

Saddlepoint approximations for subordinator processes

ABSTRACT

We develop the saddlepoint approximations in obtaining the transition functions for general subordinator processes. We derive explicit expressions of the first- and second-order approximations. Specifically, we consider some particular classes of subordinators including the Poisson processes, the Gamma processes, the α-stable subordinators, and the Poisson random integrals. We test this technique on the Poisson and Gamma processes, which have closed-form transition functions. Outcomes show that the approximate expressions are consistent with the true transition functions. We then use this method to predict transition density functions for the α-stable subordinator processes. Finally, we calculate approximated transition densities for some Poisson random integrations. Numerical analysis shows the perfect ability of the saddlepoint approximations to predict the transition densities of the α-stable processes and the Poisson random integrations.     

Alternative moment approximations for berkson's minimum logit chi-squared estimator

Expressions are derived for the bias to order J^-1 , the variance to order J^-2 and the mean squared error to order J^-2 of Berkson's minimum logit chi-squared estimator where J is the number of distinct design points. These moment approximations are numerically compared to Monte Carlo estimates of the true moments and the moment approximations of Amemiya (1980) which are appropriate when the “average” number of observations per design point is large. They are used to compare the mean squared error of the minimum logit chi-squared estimator to that of the maximum likelihood estimator and to investigate the effect of bias on confidence intenrals constructed using the minimum logit chi-squared estimator.     

Modeling longitudinal count data with dropouts

This paper explores the utility of different approaches for modeling longitudinal count data with dropouts arising from a clinical study for the treatment of actinic keratosis lesions on the face and balding scalp. A feature of these data is that as the disease for subjects on the active arm improves their data show larger dispersion compared with those on the vehicle, exhibiting an over‐dispersion relative to the Poisson distribution. After fitting the marginal (or population averaged) model using the generalized estimating equation (GEE), we note that inferences from such a model might be biased as dropouts are treatment related. Then, we consider using a weighted GEE (WGEE) where each subject's contribution to the analysis is weighted inversely by the subject's probability of dropout. Based on the model findings, we argue that the WGEE might not address the concerns about the impact of dropouts on the efficacy findings when dropouts are treatment related. As an alternative, we consider likelihood‐based inference where random effects are added to the model to allow for heterogeneity across subjects. Finally, we consider a transition model where, unlike the previous approaches that model the log‐link function of the mean response, we model the subject's actual lesion counts. This model is an extension of the Poisson autoregressive model of order 1, where the autoregressive parameter is taken to be a function of treatment as well as other covariates to induce different dispersions and correlations for the two treatment arms. We conclude with a discussion about model selection. Published in 2009 by John Wiley & Sons, Ltd.     

Additional approximations to the distribution of a structural coefficient estimator for the case of two included endogenous variables

Four new approximations t o the exact distribution of the two-stage l e a s t squares estimator of astructuralcoefficient for

the case of two included endogeneous variables are introduced and compared with the others in the literatur e . Two of the new approximations are based on the Pearson distribution and are found to be adequate throughout the parameter space. A normal approximation using exact moments and an approximation based on the saddlepoint method (Holly and Phillips,1979) are found to be

poor for a wide range of parameter values.     

On the two-parameter Bell–Touchard discrete distribution

Abstract

In this paper we introduce a new two-parameter discrete distribution which may be useful for modeling count data. Additionally, the probability mass function is very simple and it may have a zero vertex. We show that the new discrete distribution is a particular solution of a multiple Poisson process, and that it is infinitely divisible. Additionally, various structural properties of the new discrete distribution are derived. We also discuss two methods (moments and maximum likelihood) to estimate the model parameters. The usefulness of the proposed distribution is illustrated by means of real data sets to prove its versatility in practical applications.     

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.     

Win-probabilities for comparing two Poisson variables

ABSTRACT

This article considers the problem of choosing between two possible treatments which are each modeled with a Poisson distribution. Win-probabilities are defined as the probabilities that a single potential future observation from one of the treatments will be better than, or at least as good as, a potential future observation from the other treatment. Using historical data from the two treatments, it is shown how estimates and confidence intervals can be constructed for the win-probabilities. Extensions to situations with three or more treatments are also discussed. Some examples and illustrations are provided, and the relationship between this methodology and standard inference procedures on the Poisson parameters is discussed.     

Better approximate confidence intervals for a binomial parameter

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x² distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x². Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.