Series evaluation of Tweedie exponential dispersion model densities

Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to theTweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.     

Flexible Tweedie regression models for continuous data

Tweedie regression models (TRMs) provide a flexible family of distributions to deal with non-negative right-skewed data and can handle continuous data with probability mass at zero. Estimation and inference of TRMs based on the maximum likelihood (ML) method are challenged by the presence of an infinity sum in the probability function and non-trivial restrictions on the power parameter space. In this paper, we propose two approaches for fitting TRMs, namely quasi-likelihood (QML) and pseudo-likelihood (PML). We discuss their asymptotic properties and perform simulation studies to compare our methods with the ML method. We show that the QML method provides asymptotically efficient estimation for regression parameters. Simulation studies showed that the QML and PML approaches present estimates, standard errors and coverage rates similar to the ML method. Furthermore, the second-moment assumptions required by the QML and PML methods enable us to extend the TRMs to the class of quasi-TRMs in Wedderburn's style. It allows to eliminate the non-trivial restriction on the power parameter space, and thus provides a flexible regression model to deal with continuous data. We provide an R implementation and illustrate the application of TRMs using three data sets.     

Likelihood-based and Bayesian methods for Tweedie compound Poisson linear mixed models

The Tweedie compound Poisson distribution is a subclass of the exponential dispersion family with a power variance function, in which the value of the power index lies in the interval (1,2). It is well known that the Tweedie compound Poisson density function is not analytically tractable, and numerical procedures that allow the density to be accurately and fast evaluated did not appear until fairly recently. Unsurprisingly, there has been little statistical literature devoted to full maximum likelihood inference for Tweedie compound Poisson mixed models. To date, the focus has been on estimation methods in the quasi-likelihood framework. Further, Tweedie compound Poisson mixed models involve an unknown variance function, which has a significant impact on hypothesis tests and predictive uncertainty measures. The estimation of the unknown variance function is thus of independent interest in many applications. However, quasi-likelihood-based methods are not well suited to this task. This paper presents several likelihood-based inferential methods for the Tweedie compound Poisson mixed model that enable estimation of the variance function from the data. These algorithms include the likelihood approximation method, in which both the integral over the random effects and the compound Poisson density function are evaluated numerically; and the latent variable approach, in which maximum likelihood estimation is carried out via the Monte Carlo EM algorithm, without the need for approximating the density function. In addition, we derive the corresponding Markov Chain Monte Carlo algorithm for a Bayesian formulation of the mixed model. We demonstrate the use of the various methods through a numerical example, and conduct an array of simulation studies to evaluate the statistical properties of the proposed estimators.     

On generalized variance of product of powered components and multiple stable Tweedie models

A flexible family of multivariate models, named multiple stable Tweedie (MST) models, is introduced and produces generalized variance functions which are products of powered components of the mean. These MST models are built from a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are also real independent Tweedie variables, with the same dispersion parameter equal to the fixed component. In this huge family of MST models, generalized variance estimators are explicitly pointed out by maximum likelihood method and, moreover, computably presented for the uniform minimum variance and unbiased approach. The second estimator is brought from modified Lévy measures of MST which lead to some solutions of particular Monge–Ampère equations.     

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     

Bias-corrected estimators for dispersion models with dispersion covariates

In this paper we discuss bias-corrected estimators for the regression and the dispersion parameters in an extended class of dispersion models (Jørgensen, 1997b). This class extends the regular dispersion models by letting the dispersion parameter vary throughout the observations, and contains the dispersion models as particular case. General formulae for the O(n⁻¹) bias are obtained explicitly in dispersion models with dispersion covariates, which generalize previous results obtained by Botter and Cordeiro (1998), Cordeiro and McCullagh (1991), Cordeiro and Vasconcellos (1999), and Paula (1992). The practical use of the formulae is that we can derive closed-form expressions for the O(n⁻¹) biases of the maximum likelihood estimators of the regression and dispersion parameters when the information matrix has a closed-form. Various expressions for the O(n⁻¹) biases are given for special models. The formulae have advantages for numerical purposes because they require only a supplementary weighted linear regression. We also compare these bias-corrected estimators with two different estimators which are also bias-free to order O(n⁻¹) that are based on bootstrap methods. These estimators are compared by simulation.     

Multivariate stable exponential families and Tweedie scale

In this paper, we characterize the multivariate stable natural exponential families by a property of homogeneity of the cumulant function of some basis, and by a property of homogeneity of the variance function. We also extend the definition of a Tweedie scale to a finite dimensional space and we give a class of natural exponential families belonging to this scale on the space of symmetric matrices.     

Discrete M-robust designs for regression models

M-robust designs are defined and constructed for misspecified linear regression models with possibly autocorrelated errors on a discrete design space. These designs minimize the mean-squared errors if linear regression models are correct with uncorrelated errors, subject to two robust constraints which control the change of the bias and the change of variance under model departures. Simulated annealing algorithm is applied to construct M-robust designs. Examples are given to show M-robust designs and compare them with minimax robust designs.     

Goodness-of-fit tests for Pareto distribution based on a characterization and their asymptotics

In this paper we present a new characterization of the Pareto distribution and consider goodness-of-fit tests based on it. We provide an integral and Kolmogorov–Smirnov-type statistics based on U-statistics and we calculate Bahadur efficiency for various alternatives. We find locally optimal alternatives for those tests. For small sample sizes, we compare the power of those tests with some common goodness-of-fit tests.     

Nonparametric models and their estimation

Summary Nonparametric models have become more and more popular over the last two decades. One reason for their popularity is software availability, which easily allows to fit smooth but otherwise unspecified functions to data. A benefit of the models is that the functional shape of a regression function is not prespecified in advance, but determined by the data. Clearly this allows for more insight which can be interpreted on a substance matter level. This paper gives an overview of available fitting routines, commonly called smoothing procedures. Moreover, a number of extensions to classical scatterplot smoothing are discussed, with examples supporting the advantages of the routines.     

Influence diagnostics in nonlinear reproductive dispersion mixed models

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example.     

Skewness of maximum likelihood estimators in dispersion models

We introduce the dispersion models with a regression structure to extend the generalized linear models, the exponential family nonlinear models (Cordeiro and Paula, 1989) and the proper dispersion models (Jørgensen, 1997a). We provide a matrix expression for the skewness of the maximum likelihood estimators of the regression parameters in dispersion models. The formula is suitable for computer implementation and can be applied for several important submodels discussed in the literature. Expressions for the skewness of the maximum likelihood estimators of the precision and dispersion parameters are also derived. In particular, our results extend previous formulas obtained by Cordeiro and Cordeiro (2001) and Cavalcanti et al. (2009). A simulation study is performed to show the practice importance of our results.     

General mixed Poisson regression models with varying dispersion

A general class of mixed Poisson regression models is introduced. This class is based on a mixing between the Poisson distribution and a distribution belonging to the exponential family. With this, we unified some overdispersed models which have been studied separately, such as negative binomial and Poisson inverse gaussian models. We consider a regression structure for both the mean and dispersion parameters of the mixed Poisson models, thus extending, and in some cases correcting, some previous models considered in the literature. An expectation–maximization (EM) algorithm is proposed for estimation of the parameters and some diagnostic measures, based on the EM algorithm, are considered. We also obtain an explicit expression for the observed information matrix. An empirical illustration is presented in order to show the performance of our class of mixed Poisson models. This paper contains a Supplementary Material.     

Generalized Ising models and their applications

The Ising model has become more prominent in spatial statistics since its applications in image analysis, as pioneered by Besag. This paper describes three multilevel generalizations of the Ising model, including the general spin Ising model. We compare and contrast these three generalizations. In all three cases, the normalizing constant is intractable but, using the developments made by physicists, we give adequate approximations for the general spin model, together with complete expressions for the binary Ising model. We show that these approximations allow inference for the general spin model. An application to texture analysis is also given.     

Testing for varying zero-inflation and dispersion in generalized Poisson regression models

Homogeneity of dispersion parameters and zero-inflation parameters is a standard assumption in zero-inflated generalized Poisson regression (ZIGPR) models. However, this assumption may be not appropriate in some situations. This work develops a score test for varying dispersion and/or zero-inflation parameter in the ZIGPR models, and corresponding test statistics are obtained. Two numerical examples are given to illustrate our methodology, and the properties of score test statistics are investigated through Monte Carlo simulations.     

Detection of influential observations in nonlinear reproductive dispersion models

A robust case-deletion diagnostic which could avoid masking is presented for nonlinear reproductive dispersion models (NRDM) (Jorgensen, 1997) that include generalized linear models and exponential family nonlinear models as special cases. Based on the second-order approximation of log-likelihood displacement and Poon & Poon's (1999) conformal normal curvature, a measure of local influence ranging from 0 to 1 is constructed. An example illustrates application of the techniques.     

On confidence regions of semiparametric nonlinear reproductive dispersion models

This article investigates the confidence regions for semiparametric nonlinear reproductive dispersion models (SNRDMs), which is an extension of nonlinear regression models. Based on local linear estimate of nonparametric component and generalized profile likelihood estimate of parameter in SNRDMs, a modified geometric framework of Bates and Wattes is proposed. Within this geometric framework, we present three kinds of improved approximate confidence regions for the parameters and parameter subsets in terms of curvatures. The work extends the previous results of Hamilton et al. [in Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions, Ann. Statist. 10, pp. 386–393, 1982], Hamilton [in Confidence regions for parameter subset in nonlinear regression, Biometrika, 73, pp. 57–64, 1986], Wei [in On confidence regions of embedded models in regular parameter families (a geometric approch), Austral. J. Statist. 36, pp. 327–338, 1994], Tang et al. [in Confidence regions in quasi-likelihood nonlinear models: a geometric approach, J. Biomath. 15, pp. 55–64, 2000b] and Zhu et al. [in On confidence regions of semiparametric nonlinear regression models, Acta. Math. Scient. 20, pp. 68–75, 2000].