A note on rescalings of the skew-normal distribution

In this article, we show that certain skew-normal parametric statistical models are a result of rescalings of the skew normal model studied by Azzalini (1985). Using this procedure we define a class of skew-normal distributions and we study its moment, skewness and kurtosis coefficients. At the end of this article we will use this class of distribution to make some extensions of the skew-normal model.

Other skew-normal models has been studied, for example by Mukhopadhyay and Vidakovic (1995), Sahu et al. (2003) and Nadarajah and Kotz (2003).These models are product of rescalings of the skew-normal model of Azzalini (1985).
The purpose in these notes, is to show how the rescalings affects the representation of the skew-normal model, as well as provide some extensions of this rescaled model.This article is organized as follows.In section 2 we give the rescalings of the skew-normal model, some properties, examples, moments and asymmetric and kurtosis coefficients.In section 3 some extensions of this rescaled model are presented.In section 4 presents the appendix.

Rescaling
Definition 1.We will say that a random variable X has a Rescaled-Skew-Normal distribution with parameters λ, r and s, with r s > 0, that will be denoted by X ∼ RSN (λ, r, s), if its density function is given by where φ and Φ are the standard normal density and distribution functions, respectively.
Proposition 1.Let U ∼ N (0, r s ) and V ∼ N (0, 1) where U and V are independent and let X = λ where r s > 0 and λ ∈ R, then the density function f X of the random variable X is given in (7).
Proof This proof is in the appendix.2 Remark 1. a) A distribution suitable for fitting positive data is the half-normal distribution.We say that a random variable X follows a half-normal distribution with scale parameter σ if its density function is given by: with σ > 0. We denote this by writing X ∼ HN(σ).
b) We observe that in the representation we have that r s > 0, since it is the variance of the normal model.Thus, we may construct skewnormal models that are product of rescalings and it must only satisfy that r s > 0. On the other hand, we can see that the distribution of |U | is the family of rescaled half normal distribution.
Therefore, the RSN model is product of a rescaling of the SN model.Properties 1.The following properties follow immediately from Definition 1.
a) The pdf of the RSN (0, r, s) is identical to the pdf of the N (0, Such model is studied by Elal et al. (2004).If consider α = β = λ then the model corresponds to the one presented by Sahu et al. (2003) in the univariate case and, on the other hand, if α = β = 0 it corresponds to the SN(λ) model by Azzalini (1985).
This model is exhibited by Mukhopadhyay and Vidakovic (1995) in an Bayesian analysis.
x ´where σ and δ corresponds to the standard deviations of the normal models, then X ∼ RSN (λ, σ, δ) since Nadarajah and Kotz(2003) introduced such model and they called Skew-Normal-Normal distribution.

Moments and Moment Generating Function of the RSN Model
Proposition 2. Let X ∼ RSN (λ, r, s) then the moments of order n are: ) Proof This proof is in the appendix.2 Proposition 3. Let X ∼ RSN (λ, r, s) then the moment generating function is: Proof This proof is in the appendix. , and from Proposition 3 we have This completes the result.2 Remark 2. Since the RSN model is a result of a rescaling of the SN model, the asymmetry and kurtosis ranges of the RSN model are equal to the SN model, in other words, the intervals coincide with the given in (5).On the other hand, any rescaling of the random variable that has a skew-normal distribution, do no affect the result of the asymmetry and kurtosis, as well as, the expressions given in (4).If in (9) we consider r = s = 1, we obtain

Extensions of the RSN model
In this section we show extensions of the skew-normal model using the rescaled skew-normal model.
, with X and Y independent, and ´.
Proof This proof is in the appendix.2 Definition 2. Let X ∼ N (0, σ 2 ), we say that T has a Truncated-Normal distribution with parameters c, σ 2 , will be denoted by T ∼ Υ σ 2 (c), if its density function, f T (t), is given by: f with F X and f X distribution and density functions of the variable X, respectively.

Appendix
Proof of the Proposition 1: By using ra 2 + sb 2 = r 2 s and differentiating with respect to x we have

2. 3 .
Asymmetry and Kurtosis Coefficients of the RSN model Proposition 4. Let X ∼ RSN (λ, r, s) then the asymmetry and kurtosis coefficients are:

( 3 )
. The b k in the Proposition 2 are the moments of the standard normal distribution, see Johnson et al. (1995).

dv the result is obtained. 2 Remark 3 .
Proposition 5  shows that the addition of a random variable rescaled skew normal and a normal to scale is a rescaled skew normal.The Proposition 6 extends the result byHenze (1986) of the extended skew normal model.When r = s = 1 in Proposition 7, we obtain the model of the Example 2. In Bayesian context, the Corollary 1 is a result of conjugate priori.When r = s in Proposition 8, we obtain the Skew-Generalized-Normal model to scale introduced by Arellano-Valle et al. (2004).

s 2 +r 2 λ 2 E 2 ´and
[X n ] = E [(a |U | + bV ) n ] E(V k ) are thek-th moment of the standard normal random variable then the result is obtained.Proof of the Proposition 3: