Flexible Tweedie regression models for continuous data

Tweedie regression models provide a flexible family of distributions to deal with non-negative highly right-skewed data as well as symmetric and heavy tailed data and can handle continuous data with probability mass at zero. The estimation and inference of Tweedie regression models based on the maximum likelihood 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 Tweedie regression models, namely, quasi- and pseudo-likelihood. We discuss the asymptotic properties of the two approaches and perform simulation studies to compare our methods with the maximum likelihood method. In particular, we show that the quasi-likelihood method provides asymptotically efficient estimation for regression parameters. The computational implementation of the alternative methods is faster and easier than the orthodox maximum likelihood, relying on a simple Newton scoring algorithm. Simulation studies showed that the quasi- and pseudo-likelihood approaches present estimates, standard errors and coverage rates similar to the maximum likelihood method. Furthermore, the second-moment assumptions required by the quasi- and pseudo-likelihood methods enables us to extend the Tweedie regression models to the class of quasi-Tweedie regression models in the Wedderburn's style. Moreover, 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 \texttt{R} implementation and illustrate the application of Tweedie regression models using three data sets.Comment: 34 pages, 8 figure

Associated kernel discriminant analysis for multivariate mixed data

Associated kernels have been introduced to improve the classical (symmetric) continuous kernels for smoothing any functional on several kinds of supports such as bounded continuous and discrete sets. In this paper, an associated kernel for discriminant analysis with multivariate mixed variables is proposed. These variables are of three types: continuous, categorical andcount. The method consists of using a product of adapted univariate associated kernels and an estimate of the misclassication rate. A new prole version cross-validation procedure of bandwidth matrices selection is introduced for multivariate mixed data, while a classical cross-validation is used for homogeneous data sets having the same reference measures. Simulations and validation results show the relevance of the proposed method. The method has been validated on real coronary heart disease data in comparison to the classical kernel discriminant analysis

Asymptotic properties of the normalized discrete associated-kernel estimator for probability mass function

Discrete kernel smoothing is now gaining importance in nonparametric statistics. In this paper, we investigate some asymptotic properties of the normalized discrete associated-kernel estimator of a probability mass function. We show, under some regularity and non-restrictive assumptions on the associated-kernel, that the normalizing random variable converges in mean square to 1. We then derive the consistency and the asymptotic normality of the proposed estimator. Various families of discrete kernels already exhibited satisfy the conditions, including the refined CoM-Poisson which is underdispersed and of second-order. Finally, the first-order binomial kernel is discussed and, surprisingly, its normalized estimator has a suitable asymptotic behaviour through simulations.Comment: 20 pages, 3 figure

On semiparametric regression for count explanatory variables

International audienceWe study the problem of semiparametric estimation of a multivariate count regression function m : Nd -> R that can be represented as a product of an unknown discrete parametric function r and an unknown discrete smooth function w. For the construction of such estimators, we first find an approximation result br for the parametric part r, and then estimate the nonparametric multiplicative correction factor w = m/br by a discrete associated-kernel method. Comparisons are therefore carried out with the nonparametric count regression estimator of Nadaraya-Watson type. We point out that the new semiparametric count regression estimator can reduce the bias with respect to purely nonparametric count regression estimator, without affecting the variance

Some discrete exponential dispersion models : poisson-tweedie and hinde-demétrio classes

In this paper we investigate two classes of exponential dispersion models (EDMs) for overdispersed count data with respect to the Poisson distribution. The first is a class of Poisson mixture with positive Tweedie mixing distributions. As an approximation (in terms of unit variance function) of the first, the second is a new class of EDMs characterized by their unit variance functions of the form µ + µp, where p is a real index related to a precise model. These two classes provide some alternatives to the negative binomial distribution (p = 2) which is classically used in the framework of regression models for count data when overdispersion results in a lack of fit of the Poisson regression model. Some properties are then studied and the practical usefulness is also discussed

On d-pseudo-orthogonality of the Sheffer systems associated to a convolution semigroup

We investigate which Sheffer polynomials can be associated to a convolution semigroup of probability measures, usually induced by a stochastic process with stationary and independent increments. From a recent notion of d-pseudo-orthogonality (d ? {2, 3, ? ? ?}), we characterize the associated d-pseudo-orthogonal polynomials by the class of generating probability measures, which belongs to the natural exponential family with polynomial variance functions of exact degree 2d - 1. This extends some results of (classical) orthogonality; in particular, some new sets of martingales are then pointed out. For each integer d = 2 we completely illustrate polynomials with (2d-1)-term recurrence relation for the families of positive stable processes