On multiple constraint skewed models

Abstract

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.

Flexible Birnbaum–Saunders Distribution

“…Based on the flexible skew-normal model proposed in [2], we extend the Birnbaum-Saunders. The main idea is to apply (7) with Z ∼ FSN(δ, λ) introduced in (4). This new model is called the flexible Birnbaum-Saunders (FBS) distribution whose pdf is given by…”

“…where φ and Φ are the pdf and cdf of the N(0, 1) distribution, respectively, and c −1 δ = 1 − Φ(δ). Other recent proposals in the contemporary literature dealing with bimodality are the extended two-pieces skew-normal model (ETN), introduced in [7] and the uni-bi-modal asymmetric power normal model given in [8] whose properties are based on results given in [9,10]. Applications of interest in Economics are given in [11].…”

Multiple constraint and truncated skew models

“…[3] Consequently, the moments of the 0-truncated skew-normal distribution (13) are given by E[X r ] = σ r E[Z r ], so that Simulation of STN(σ , α) variables may be accomplished using the quantile function of the skew-normal distribution. Thus if U ∼ U(0, 1), then defining,…”

“…, g k which are not necessarily linear. Arnold et al [3] provide discussion of a relatively simple two constraint model of this more general form,…”

Likelihood-Based Inference for the Asymmetric Beta-Skew Alpha-Power Distribution

“…where φ(•) and Φ(•) are the pdf and cdf of the standard normal distribution, respectively. Among the densities of bimodal type supported under general structure (1), it can be stood out the proposals of Arnold et al [2], Gómez et al [3] and Kim [4]. Based on the alpha-power family of distributions by Durrans [5] with pdf given by…”