An extension of the skew-generalized normal distribution and its derivation

In this paper, we introduce a new class of skew-symmetric distributions which are formulated based on cumulative distributions of skew-symmetric densities. This new class is an extension of other skew-symmetric distributions that have already been studied. We give special attention to a family from this class that could be seen as an extension of the skew-generalized-normal model introduced by ArellanoValle et al.(2004). We study the main properties, stochastic representation, moments and an extension of this new model.


Introduction
It is known that the normal distribution is very useful in different applications in statistics.However, it is also known that this distribution is not appropriate in those cases where symmetry is absent.In this respect the skew-normal family has been studied among others by Azzalini (1985), Henze (1986), Pewsey (2000) and Martinez et al. (2008), and reviewed in the book by Genton (2004).The univariate skew-normal model has the following density function: where φ(x) and Φ(x) denote the density function and cumulative distribution function of the standard normal distribution, respectively.We denote the distribution of Azzalini as X ∼ SN(µ, σ, λ).A better family of distributions that shows a better behavior, particularly on the side with smaller mass, than the skew-normal distribution is the Skew-Generalized Normal (SGN) Distribution introduced by Arellano-Valle et al. (2004), which has the following density function: We denote this distribution by X ∼ SGN (µ, σ, λ 1 , λ 2 ).The main objective of this paper is to introduce a new class of asymmetric distributions that contain an unlimited number of skew-symmetric models.We give special attention to a skew model that is an extension of the skew-generalized normal distribution and the skew normal distribution.In Section 2, we define this new extension and call it the Extended Skew Generalized Normal Distribution (ESGN), and also study its basic properties.In Section 3, we develop important probabilistic properties of the new family of distribution, including the stochastic representation.In Section 4, we obtain the moments of the random variable with ESGN distribution and finally, Section 5 deals with a location-scale extension of the ESGN distribution and derives a generalization of the ESGN distribution.

A Class of Skew Distribution
The following Lemma was presented by Azzalini (2005), and is a fundamental result for generating skew-symmetric distributions.
Lemma 1.Let g be a probability density function symmetric about zero, and G is a distribution function such that G 0 there exist and is a density function symmetric about zero, then is a density function for any odd function w(•).Some subclass of skew-symmetric distributions generated from this lemma have been studied among others by Gupta et al. (2002), Nadarajah and Kotz (2003) and Gómez et al. (2007).In the following proposition we introduce a new subclass of skew-symmetric models.
Proposition 1.Let g be a probability density function symmetric about zero, and H n(x) be a distribution function, from a skew density function, (3) where n(x) is any odd function, then is a density function for any odd function m(•).
Now, since n(x) and m(x) are odd functions, we have , which shows that R 0 is symmetric about zero.Thus, by using Lemma 1 we prove this proposition.2 Example 1.If we consider H n(x) (m(x)) to be an extension of the cumulative distribution function from the symmetric density function g, then we obtain an extension of the family studied by Gupta et al. (2002).
Example 2. If we consider H n(x) (m(x)) to be the cumulative distribution function of a skew-symmetric density and g = φ, then we obtain an extension of the family studied by Nadarajah and Kotz (2003).
) and g to be any symmetric density function, then we obtain an extension of the family studied by Gómez et al. (2007).
The following example is the main interest of this paper and we present it as a definition.Definition 1.A random variable X is said to have a Extended Skew Generalized Normal Distribution if its density function is given by where We denote the Extended Skew Generalized Normal Distribution as x 2 and by applying Proposition 1 we can prove that (5) is a density function.
Proof: The proofs of the items (a) and (b) are obtained directly by using the functions given in ( 5) and ( 6) of Definition 1. 2 Remark 3. When λ = 0 in part 4 of Proposition 2(b), we have that: where Φ λ 1 ,λ 2 (x) is the CDF of the Skew-Generalized Normal Distribution.

Some Important Properties
We derive important probabilistic properties from the ESGN distribution.The following two propositions show similar results to those obtained from the SGN distribution, where the distribution of the absolute value of a ESGN distribution is half-normal, and the square of the ESGN distribution is a chi-square with one degree of freedom.The Proposition 4 in this section shows the stochastic representation of the ESGN distribution.Proof: Let W = |X|, then the density function of W is given by: which has the same distribution as |Y |. 2 Proof: , by using the expression in (2.6), where µ(x, λ 2 , λ)

Moments
In this section we obtain the n-th moment of the random variable X ∼ ESGN (λ 1 , λ 2 , λ).In the case that n is even, the moments are derived from the chi-square distribution with 1 degree of freedom.When n is odd, we can derive an implicit expression for the n-th moment of X.In this respect, we denote: We can show that for all λ 2 ≥ 0: Proposition 5. Let X be a random variable with X ∼ ESGN (λ 1 , λ 2 , λ).
Remark 4. Note that the distribution ( 11) is a model with five parameters, estimating they can be a problem.But today, due to computing resources, particularly the statistical software, are becoming increasingly efficient, we believe it is necessary support when desired perform this task.
Definition 3. The random variable X with density function An extension of the skew-generalized normal distribution and ...
has a Beyond Skew-Generalized Normal (BSGN) Distribution, which is a generalization of the random variable with the density function given in (5).We denote this generalization as X ∼ BSGN (α, λ 1 , λ 2 , λ).
1. We note that for α = 0 we reproduce the Extended Skew Generalized Normal Distribution.

We may consider
x 2 and by applying Proposition 1 we can prove that (12) is a density function.Proof: