Statistical Modelling

When count data exhibit excess zero, that is more zero counts than a simpler parametric distribution can model, the zero-inflated Poisson (ZIP) or zero-inflated negative binomial (ZINB) models are often used. Variable selection for these models is even more challenging than for other regression situations because the availability of p covariates implies 4 ^p possible models. We adapt to zero-inflated models an approach for variable selection that avoids the screening of all possible models. This approach is based on a stochastic search through the space of all possible models, which generates a chain of interesting models. As an additional novelty, we propose three ways of extracting information from this rich chain and we compare them in two simulation studies, where we also contrast our approach with regularization (penalized) techniques available in the literature. The analysis of a typical dataset that has motivated our research is also presented, before concluding with some recommendations.

We propose a new class of discrete generalized linear models based on the class of Poisson–Tweedie factorial dispersion models with variance of the form μ + ϕ μ p , where μ is the mean and ϕ and p are the dispersion and Tweedie power parameters, respectively. The models are fitted by using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions for the estimation of the regression and dispersion parameters, respectively. This provides a flexible and efficient regression methodology for a comprehensive family of count models including Hermite, Neyman Type A, Pólya–Aeppli, negative binomial and Poisson-inverse Gaussian. The estimating function approach allows us to extend the Poisson–Tweedie distributions to deal with underdispersed count data by allowing negative values for the dispersion parameter ϕ . Furthermore, the Poisson–Tweedie family can automatically adapt to highly skewed count data with excessive zeros, without the need to introduce zero-inflated or hurdle components, by the simple estimation of the power parameter. Thus, the proposed models offer a unified framework to deal with under-, equi-, overdispersed, zero-inflated and heavy-tailed count data. The computational implementation of the proposed models is fast, relying only on a simple Newton scoring algorithm. Simulation studies showed that the estimating function approach provides unbiased and consistent estimators for both regression and dispersion parameters. We highlight the ability of the Poisson–Tweedie distributions to deal with count data through a consideration of dispersion, zero-inflated and heavy tail indices, and illustrate its application with four data analyses. We provide an R implementation and the datasets as supplementary materials.

This article introduces a robust extension of the mixture of factor analysis models based on the restricted multivariate skew-t distribution, called mixtures of skew-t factor analysis (MSTFA) model. This model can be viewed as a powerful tool for model-based clustering of high-dimensional data where observations in each cluster exhibit non-normal features such as heavy-tailed noises and extreme skewness. Missing values may be frequently present due to the incomplete collection of data. A computationally feasible EM-type algorithm is developed to carry out maximum likelihood estimation and create single imputation of possible missing values under a missing at random mechanism. The numbers of factors and mixture components are determined via penalized likelihood criteria. The utility of our proposed methodology is illustrated through analysing both simulated and real datasets. Numerical results are shown to perform favourably compared to existing approaches.

In this work, we propose a Bayesian quantile regression method to response variables with mixed discrete-continuous distribution with a point mass at zero, where these observations are believed to be left censored or true zeros. We combine the information provided by the quantile regression analysis to present a more complete description of the probability of being censored given that the observed value is equal to zero, while also studying the conditional quantiles of the continuous part. We build up a Markov Chain Monte Carlo method from related models in the literature to obtain samples from the posterior distribution. We demonstrate the suitability of the model to analyse this censoring probability with a simulated example and two applications with real data. The first is a well-known dataset from the econometrics literature about women labour in Britain, and the second considers the statistical analysis of expenditures with durable goods, considering information from Brazil.