This paper considers an extension to the skew-normal model through the inclusion of an additional parameter which can lead to both uni-and bi-modal distributions. The paper presents various basic properties of this family of distributions and provides a stochastic representation which is useful for obtaining theoretical properties and to simulate from the distribution. Moreover, the singularity of the Fisher information matrix is investigated and maximum likelihood estimation for a random sample with no covariates is considered. The main motivation is thus to avoid using mixtures in fitting bimodal data as these are well known to be complicated to deal with, particularly because of identifiability problems. Data-based illustrations show that such model can be useful.Environmetrics is a mixture of the skew-normal distributions in the asymmetry parameter. With more general models in view, Azzalini (1985) introduces a fundamental result for the development of skew-symmetric models which is presented in the following. Lemma 1. Let f 0 be a probability density function symmetric about zero, and G is a distribution function such that G exists and is a density function symmetric about zero, thenis a density function for any α ∈ R. We use the notation Z ∼ Sf 0 (α) to denote the fact that Z is distributed according to the density above.Using the result in this lemma it is possible to generate a wide range of asymmetric families of unimodal and bimodal skew distributions. We have, for example, the unimodal family considered by Gupta et al. (2002) where f 0 is taken as the Laplace, logistic and uniform densities and G their respective distribution functions. Similar constructions has been implemented in some other papers such as Balakrishann and Ambagaspitiya (1994) and Arnold and Beaver (2000a,b). Nadarajah and Kotz (2003) consider f 0 ≡ φ fixed and G as the cumulative distribution of the normal, Student-t, Cauchy, Laplace, logistic and uniform, respectively. Gómez et al. (2007) consider f 0 to be any symmetric density and G ≡ fixed. However, recently some authors such as as Azzalini and are exploring models asymmetric bimodal. Using mixtures of skew distributions, models for fitting multimodal data can be found in , using mixtures of skew-normal distributions and in using mixtures of skew-t distributions, where data sets with bimodal features are also presented. Typically, the models proposed are somewhat involved in terms of computational implementation and they present identifiability (label switching) problems.The principal interest of the present paper is to study properties and inferential aspects of a new bimodal family which is generated from the construction discussed for the skew distributions in Lemma 1. Motivation for considering such family of models is the lacking in the literature of parametric models that can more adequately replace mixtures of distributions which, as is quite well known, presents serious estimation problems either from classical or Bayesian approaches (McLachan and Peel, 2000;Marin et al....

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.

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