Hugo S. Salinas

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.

On multiple constraint skewed models

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.

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.

Bimodal Symmetric-Asymmetric Power-Normal Families

ABSTRACT This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005) and Arnold et al. (2009) to the exponentiated normal distribution studied in Durrans (1992), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted by BPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations.

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.

Properties and Inference on the Skew-Curved-Symmetric Family of Distributions

In this article, we study some results related to a specific class of distributions, called skew-curved-symmetric family of distributions that depends on a parameter controlling the skewness and kurtosis at the same time. Special elements of this family which are studied include symmetric and well-known asymmetric distributions. General results are given for the score function and the observed information matrix. It is shown that the observed information matrix is always singular for some special cases. We illustrate the flexibility of this class of distributions with an application to a real dataset on characteristics of Australian athletes.

INFORMATION MATRIX FOR GENERALIZED SKEW - NORMAL DISTRIBUTIONS

The Fisher information matrix for Generalized skew-normal (GSN) distribution is derived. The expressions for the elements of the matrices require of integrals that are solved numerically using a suitable software.

Bimodality based on the generalized skew-normal distribution

ABSTRACT This paper focuses on the development of a new extension of the generalized skew-normal distribution introduced in Gómez et al. [Generalized skew-normal models: properties and inference. Statistics. 2006;40(6):495–505]. To produce the generalization a new parameter is introduced, the signal of which has the flexibility of yielding unimodal as well as bimodal distributions. We study its properties, derive a stochastic representation and state some expressions that facilitate moments derivation. Maximum likelihood is implemented via the EM algorithm which is based on the stochastic representation derived. We show that the Fisher information matrix is singular and discuss ways of getting round this problem. An illustration using real data reveals that the model can capture well special data features such as bimodality and asymmetry.

Skew-normal alpha-power model [Statistics 48(2014) 1414–1428]

ABSTRACT In [Martínez-Flórez G, Bolfarine H, Gómez HW. Skew-normal alpha-power model. Statistics. 2014;48(6):1414–1428] the authors make an error in calculating the information matrix of the skew-normal alpha-power model under the normality hypothesis . Here we prove that the information matrix under is singular and we proposed a reparametrization that allows one to calculate the asymptotic distribution of the maximum likelihood estimator.

Generalized skew-normal models: properties and inference [Statistics 40 (2006) 495–505]

In [H.W. Gómez, H.S. Salinas and H. Bolfarine, Generalized skew-normal models: Properties and inference, Statistics 40(6) (2006), pp. 495–505] introduces a new family of asymmetric distributions that depends on two parameters called, α and β, such as for the particular case β = 0 obtained skew-normal distribution. In this note we give a corrected version for the expression that is used in calculating the moments of such distribution.