Linear models with an inverse gaussian poisson error distribution

The inverse Gaussian-Poisson (two-parameter Sichel) distribution is useful in fitting overdispersed count data. We consider linear models on the mean of a response variable, where the response is in the form of counts exhibiting extra-Poisson variation, and assume an IGP error distribution. We show how maximum likelihood estimation may be carried out using iterative Newton-Raphson IRLS fitting, where GLIM is used for the IRLS part of the maximization. Approximate likelihood ratio tests are given.     

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.     

Multivariate geometric distributions

Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping roles for different counts, A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Goldberg (1934), Bates and Meyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppulori (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation is provided.     

Prediction from a normal model using a generalized inverse Gaussian prior

In this paper, we derive prediction distribution of future response(s) from the normal distribution assuming a generalized inverse Gaussian (GIG) prior density for the variance. The GIG includes as special cases the inverse Gaussian, the inverted chi-squared and gamma distributions. The results lead to Bessel-type prediction distributions which is in contrast with the Student-t distributions usually obtained using the inverted chi-squared prior density for the variance. Further, the general structure of GIG provides us with new flexible prediction distributions which include as special cases most of the earlier results obtained under normal-inverted chi-squared or vague priors.     

Finite mixtures of natural exponential families

Let μ be a positive measure concentrated on R⁺ generating a natural exponential family (NEF) F with quadratic variance function V_F(m), m being the mean parameter of F. It is shown that v(dx) = (γ+x)μ(γ ≥ 0) (γ ≥ 0) generates a NEF G whose variance function is of the form l(m)Δ+cΔ(m), where l(m) is an affine function of m, Δ(m) is a polynomial in m (the mean of G) of degree 2, and c is a constant. The family G turns out to be a finite mixture of F and its length-biased family. We also examine the cases when F has cubic variance function and show that for suitable choices of γ the family G has variance function of the form P(m) + Q(m)m where P, Q are polynomials in m of degree m2 while Δ is an affine function of m. Finally we extend the idea to two dimensions by considering a bivariate Poisson and bivariate gamma mixture distribution.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Discrete distributions and purchasing models

Ihis paper reviews the use of different discrete distributions in the modelling of consumer purchasing behaviour.. A feature of the work is the extensive empirical validation of the models. Some interesting characterizations are briefly discussed.. Ihe paper concludes with some unresolved problems and suggested areas for future research     

On families of beta- and generalized gamma-generated distributions and associated inference

A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

Entropy estimation and goodness-of-fit tests for the inverse Gaussian and Laplace distributions using paired ranked set sampling

ABSTRACT

In this paper, Vasicek [A test for normality based on sample entropy. J R Stat Soc Ser B. 1976;38:54–59] entropy estimator is modified using paired ranked set sampling (PRSS) method. Also, two goodness-of-fit tests using PRSS are suggested for the inverse Gaussian and Laplace distributions. The new suggested entropy estimator and goodness-of-fit tests using PRSS are compared with their counterparts using simple random sampling (SRS) via Monte Carlo simulations. The critical values of the suggested tests are obtained, and the powers of the tests based on several alternatives hypotheses using SRS and PRSS are calculated. It turns out that the proposed PRSS entropy estimator is more efficient than the SRS counterpart in terms of root mean square error. Also, the proposed PRSS goodness-of-fit tests have higher powers than their counterparts using SRS for all alternative considered in this study.     

Negative variance components for non-negative hierarchical data with correlation,over-, and/or underdispersion

The concept of negative variance components in linear mixed-effects models, while confusing at first sight, has received considerable attention in the literature, for well over half a century, following the early work of Chernoff [7 H. Chernoff, On the distribution of the likelihood ratio, Ann. Math. Statist. 25 (1954), pp. 573–578.[Crossref] , [Google Scholar]] and Nelder [21 J.A. Nelder, The interpretation of negative components of variance, Biometrika 41 (1954), pp. 544–548.[Crossref], [Web of Science ®] , [Google Scholar]]. Broadly, negative variance components in linear mixed models are allowable if inferences are restricted to the implied marginal model. When a hierarchical view-point is adopted, in the sense that outcomes are specified conditionally upon random effects, the variance–covariance matrix of the random effects must be positive-definite (positive-semi-definite is also possible, but raises issues of degenerate distributions). Many contemporary software packages allow for this distinction. Less work has been done for generalized linear mixed models. Here, we study such models, with extension to allow for overdispersion, for non-negative outcomes (counts). Using a study of trichomes counts on tomato plants, it is illustrated how such negative variance components play a natural role in modeling both the correlation between repeated measures on the same experimental unit and over- or underdispersion.     

Random effects regression models for count data with excess zeros in caries research

We extend the family of Poisson and negative binomial models to derive the joint distribution of clustered count outcomes with extra zeros. Two random effects models are formulated. The first model assumes a shared random effects term between the conditional probability of perfect zeros and the conditional mean of the imperfect state. The second formulation relaxes the shared random effects assumption by relating the conditional probability of perfect zeros and the conditional mean of the imperfect state to two different but correlated random effects variables. Under the conditional independence and the missing data at random assumption, a direct optimization of the marginal likelihood and an EM algorithm are proposed to fit the proposed models. Our proposed models are fitted to dental caries counts of children under the age of six in the city of Detroit.