Héctor W. Gómez

A New Class of Skew-Normal Distributions

Abstract We introduce a new family of asymmetric normal distributions that contains Azzalinis skew-normal (SN) distribution as a special case. We study the main properties of this new family, showing in particular that it may be generated via mixtures on the SN asymmetry parameter when the mixing distribution is normal. This property provides a Bayesian interpretation of the new family.

Bayesian inference for shape mixtures of skewed distributions, with application to regression analysis

We introduce a class of shape mixtures of skewed distributions and study some of its main properties. We discuss a Bayesian interpretation and some invariance results of the proposed class. We develop a Bayesian analysis of the skew-normal, skew-generalized-normal, skew-normal-t and skew-t-normal linear re- gression models under some special prior specications for the model parameters. In particular, we show that the full posterior of the skew-normal regression model parameters is proper under an arbitrary proper prior for the shape parameter and noninformative prior for the other parameters. We implement a convenient hierar- chical representation in order to obtain the corresponding posterior analysis. We illustrate our approach with an application to a real dataset on characteristics of Australian male athletes.

Large-Sample Inference for the Epsilon-Skew-t Distribution

The main object of this article is to discuss maximum likelihood inference for the epsilon-skew-t distribution. Special cases of this distribution include the epsilon-skew-Cauchy and the epsilon-skew-normal distributions. We derive the information matrix for the maximum likelihood estimators. The approach is applied to a data set presenting significant amount of skewness and heavy tails. In the application we consider the epsilon-skew-t distribution with known and unknown degrees of freedom parameter, showing great flexibility in adjusting to skew data with heavy tails.

An Extension of the Epsilon-Skew-Normal Distribution

This article is related with the probabilistic and statistical properties of an parametric extension of the so-called epsilon-skew-normal (ESN) distribution introduced by Mudholkar and Hutson (2000), which considers an additional shape parameter in order to increase the flexibility of the ESN distribution. Also, this article concerns likelihood inference about the parameters of the new class. In particular, the information matrix of the maximum likelihood estimators is obtained, showing that it is non singular in the special normal case. Finally, the statistical methods are illustrated with two examples based on real datasets.

Generalized skew-normal models: properties and inference

In this article, we introduce a new family of asymmetric distributions, which depends on two parameters namely, α and β, and in the special case where β = 0, the skew-normal (SN) distribution considered by Azzallini [Azzalini, A., 1985, A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.] is obtained. Basic properties such as a stochastic representation and the derivation of maximum likelihood and moment estimators are studied. The asymptotic behaviour of both types of estimators is also investigated. Results of a small-scale simulation study is provided illustrating the usefulness of the new model. An application to a real data set is reported showing that it can present better fit than the SN distribution.

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.

Modified slash distribution

In this paper we introduce a modified slash distribution obtained by modifying the usual slash distribution. This new distribution is based on the quotient of two independent random variables, whose distributions are the normal and the power of an exponential distribution of scale parameter equals to two, respectively. In this way, the result is a new distribution whose kurtosis values are greater when compared with that of the slash distribution. We study the density, some properties, moments, kurtosis and make inferences by the method of moments and maximum likelihood. We introduce a multivariate version of this new distribution. Moreover, we provide two illustrations with real data showing that the new distribution fits better the data than the ordinary slash distribution.

Asymmetric regression models with limited responses with an application to antibody response to vaccine.

We develop regression models for limited and censored data based on the mixture between the log-power-normal and Bernoulli-type distributions. A likelihood-based approach is implemented for parameter estimation and a small-scale simulation study is conducted to evaluate parameter recovery, with emphasis on bias estimation. The main conclusion is that the approach is very much satisfactory for moderate and large sample sizes. A real data example, the safety and immunogenecity study of measles vaccine in Haiti, is presented to illustrate how different models can be used to fit this type of data. As shown, the asymmetric models considered seem to present the best fit for the data set under study, revealing significance of the explanatory variable sex, which is not found significant with the log-normal model.

A doubly skewed normal distribution

We consider a distribution obtained by combining two well-known mechanisms for generating skewed distributions. In this manner we arrive at a flexible model which subsumes and extends several skew distributions which have been discussed in the literature. One approach to the problem of generating skewed distributions was first popularized by Azzalini [A class of distributions which includes the normal ones. Scand J Stat. 1985;12:171–178]. The single constraint skew normal distribution that was studied by Azzalini is of the form where φ and Φ denote, respectively, the standard normal density and distribution function and α∈ℝ is a skewing parameter. Multiple constraint variations of this distribution have also been considered. The second skewing approach that we will consider was proposed by Mudholkar and Hutson [The epsilon-skew-normal distribution for analyzing near-normal data. J Statist Plann Inference. 2000;83:291–309] and called an epsilon-skew-normal distribution. The combination of an Azzalini mechanism with that of Mudholkar and Hutson is investigated in this paper with special focus on the distributions obtained using the standard normal as the base distribution. The resulting flexible model includes both unimodal and bimodal cases and can be expected to fit a wider variety of data configurations than either of the models involving a single skewing mechanism. Distributional and inferential properties of the doubly skewed model are discussed and the model is used to obtain improved fits to two well-known data sets.

Epsilon half-normal model: Properties and inference

The half-normal distribution is one of the widely used probability distribution for non-negative data modeling, specifically, to describe the lifetime process under fatigue. In this paper, we introduce a new type of non-negative distribution that extends the half-normal distribution. We refer to this new distribution as the epsilon half-normal distribution. We provide mathematical properties of this new distribution. In particular, we derive the stochastic representation, explicit formulas for the n-th moment, the asymmetry and kurtosis coefficients and the moment generating function. We also discuss some inferential aspects related to the maximum likelihood estimation. We illustrate the flexibility of this type of distribution with an application to a real dataset of stress-rupture.