Variance- mean mixture of CSN X. Zhu Variance-mean mixture of closed skew Introduction of skew normal distributions normal distributions Variance- mean mixture of CSN distributions Dr. Xiaonan Zhu References Department of Mathematics University of North Alabama [email protected] October 10, 2020 X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 1 / 25 Outline Variance- mean mixture of CSN X. Zhu Introduction of skew normal distributions 1 Introduction of skew normal distributions Variance- mean mixture of CSN distributions References 2 Variance-mean mixture of CSN distributions X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 2 / 25 Variance- mean mixture of CSN X. Zhu Introduction of skew normal distributions Variance- mean mixture 1 Introduction of skew normal distributions of CSN distributions References X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 3 / 25 Introduction of skew normal distributions Variance- mean mixture In practical applications, skew data sets occur in many of CSN X. Zhu fields, such as economics, finance, biomedicine, etc. Introduction of skew normal distributions Variance- mean mixture "Normality is a myth; there never was, and never of CSN distributions will be, a normal distribution." References –R. C. Geary and E. S. Pearson, Tests of normality, Biometrika Office, University Colle, 1938 X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 4 / 25 Variance- mean mixture of CSN X. Zhu Introduction of skew normal distributions Variance- mean mixture of CSN distributions References Figure 1.1: House Prices in Cambridge, UK, 2014 http://www.cutsquash.com/2015/02/visualising-house-price-data-ipython/ X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 5 / 25 The skew normal distribution Variance- mean mixture To describe skew data sets, the skew normal distribution of CSN X. Zhu was defined by Azzalini (1985). Introduction of Definition skew normal distributions A random variable X is said to have a skew normal Variance- µ σ mean mixture distribution with location parameter , scale parameter of CSN and skewness parameter α, denoted by X ∼ SN(µ, σ2; α), if distributions References its density function is 2 x − µ x − µ f (x) = φ Φ α ; σ σ σ where µ 2 R, σ > 0, α 2 R, φ and Φ are p.d.f. and c.d.f. of N(0; 1), resp. If µ = 0 and σ = 1, then X ∼ SN(α) is called a standard skew normal random variable with skewness α. X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 6 / 25 The skew normal distribution Variance- mean mixture of CSN X. Zhu Introduction of skew normal distributions Variance- mean mixture of CSN distributions References Figure 1.2: Densities of SN(α). X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 7 / 25 Why we use SN? Variance- mean mixture Nice properties of SN(µ, σ2; α): of CSN X. Zhu The density of SN(µ, σ2; 0) is identical to the density of 2 Introduction of N(µ, σ ). skew normal distributions Variance- If Z ∼ SN(α) and Z0 ∼ N(0; 1), then jZ j and jZ0j have mean mixture of CSN the same density. distributions 2 2 References If Z ∼ SN(α), then Z ∼ χ1, the chi-square distribution with 1 degree of freedom. Simple forms of density, moments, c.d.f., etc. X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 8 / 25 Remark Components of X are always dependent when α 6= 0. The multivariate skew normal distribution Variance- mean mixture Azzalini and Dalla Valle (1996) and Azzalini and Capitanio of CSN X. Zhu (1999) extended skew normal distributions to multivariate cases. Introduction of skew normal distributions Definition Variance- A random vector X follows a multivariate skew-normal mean mixture of CSN distribution, denoted by X ∼ SNn (µ; Σ; α), if its density distributions function is References 0 −1=2 n f (x; µ; Σ; α) = 2φn (x; µ; Σ)Φ(α Σ (x − µ)); x 2 R : X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 9 / 25 The multivariate skew normal distribution Variance- mean mixture Azzalini and Dalla Valle (1996) and Azzalini and Capitanio of CSN X. Zhu (1999) extended skew normal distributions to multivariate cases. Introduction of skew normal distributions Definition Variance- A random vector X follows a multivariate skew-normal mean mixture of CSN distribution, denoted by X ∼ SNn (µ; Σ; α), if its density distributions function is References 0 −1=2 n f (x; µ; Σ; α) = 2φn (x; µ; Σ)Φ(α Σ (x − µ)); x 2 R : Remark Components of X are always dependent when α 6= 0. X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 9 / 25 Variance- mean mixture of CSN Example 0 X. Zhu Let X = (X1; ··· ; Xn) be a dependent sample such that Introduction of X ∼ SNn(0n; In; α1n) with α 6= 0; where In is the identity skew normal 0 n distributions matrix and 1n = (1; ··· ; 1) 2 R . Variance- mean mixture of CSN It can be shown that X1; ··· ; Xn are identically distributed as distributions α References Xi ∼ SN 0; 1; p ; i = 1; ··· ; n: 1+(n−1)α2 Note that p α ! 0 as n ! 1. So it is not reasonable 1+(n−1)α2 to use SNn to model a skewed population when the sample size is large. X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 10 / 25 Skew normal mixture distributions Variance- mean mixture In 2014, Vilca et al. (Vilca et al., 2014) defined the of CSN X. Zhu multivariate Skew-Normal Generalized Hyperbolic (SNGH) distribution by using the stochastic representation Introduction of skew normal distributions X = µ + W Z; (1) Variance- mean mixture p of CSN where µ 2 , W ∼ GIG(λ, χ, ), and Z follows a distributions R multivariate skew normal (SN) distribution, denoted by References Z ∼ SNp(0; Σ; α), with the density function 0 − 1 p f (z) = 2φp(z; Σ)Φ(α Σ 2 z); z 2 R : X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 11 / 25 Skew normal mixture distributions Variance- mean mixture In 2015, Arslan (Arslan, 2015) studied a more general case, of CSN X. Zhu the variance-mean mixture of the multivariate SN distribution, defined by Introduction of skew normal 1 1 distributions −1 − X = µ + W γ + W 2 Σ 2 Z; (2) Variance- mean mixture of CSN p 1 distributions where µ; γ 2 R , Σ 2 is the square root of a positive definite References matrix Σ 2 Mp×p, Z ∼ SNp(0; Ip; α) and W ∼ GIG(λ, χ, ). X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 12 / 25 Variance- mean mixture of CSN X. Zhu Introduction of skew normal distributions Variance- mean mixture 2 Variance-mean mixture of CSN distributions of CSN distributions References X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 13 / 25 Variance- mean mixture The closed skew normal distribution was introduced by of CSN González-Farías et al. (2004). X. Zhu Introduction of Definition skew normal distributions An n-dimensional random vector X is distributed according Variance- to a closed skew normal distribution with parameters mean mixture of CSN m; µ; Σ; D and ∆, denoted by X ∼ CSNn;m(µ; Σ; D; ∆), if its distributions density is References fn;m(x; µ; Σ; D; ∆) = Cφn(x; µ; Σ)Φm[D(x − µ); 0m; ∆]; −1 0 where C = Φm(0; 0; ∆ + DΣD ) and Φm(·; 0m; ∆) is the m-dimensional normal distribution function with mean vector 0m and covariance matrix ∆, X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 14 / 25 Variance-mean mixture of CSN distributions Variance- mean mixture of CSN Definition X. Zhu A p-variate random vector X is said to have a Introduction of skew normal variance-mean mixture of the CSN distribution if it has a distributions following stochastic representation, Variance- mean mixture of CSN 1 distributions X = µ + W γ + W 2 Z; (3) References where Z ∼ CSNp;m(0; Σ; D; ∆) is independent of W ∼ GIG(λ, χ, ), µ; γ 2 Rp, denoted by X ∼ VMMCSNp;m(θ), θ = (λ, χ, ; µ; γ; Σ; D; ∆). X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 15 / 25 Variance- mean mixture of CSN Theorem X. Zhu Let X ∼ VMMCSNp;m(θ). The moment generating function Introduction of of X is given by skew normal distributions exp(t0µ) 1 Variance- 0 0 MX(t) = 0 MGIG t γ + t Σt F1(0); mean mixture Φm(0; ∆ + DΣD ) 2 of CSN distributions p 0 0 References for t 2 R such that − 2t γ − t Σt > 0, where MGIG(·) is the moment generating function of GIG(λ, χ, ) and F1(·) is the cumulative distribution function of the p-dimensional GH (λ, χ, − 2t0γ − t0Σt; 0; ∆ + DΣD0; −DΣt). X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 16 / 25 Variance- mean mixture of CSN Theorem X. Zhu The density function of X ∼ VMMCSNp;m(θ) is given by Introduction of skew normal 1 distributions f (x) = f (x)F (D(x − µ)) (4) Φ (0; ∆ + DΣD0) GH 2 Variance- m mean mixture of CSN distributions where F2(·) is the cumulative distribution function of the References m-dimensional multivariate GH distribution p 0 −1 GH λ − 2 ; s + χ, γ Σ γ + ; 0; ∆; Dγ , 0 −1 s = (x − µ) Σ (x − µ) and fGH (·) is the density function of the GH distribution. X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 17 / 25 Example Variance- mean mixture To illustrate how the distribution of X ∼ VMMCSNp;m(θ) of CSN X.