Variance-Mean Mixture of Closed Skew Normal Distributions

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 ∈ R, σ > 0, α ∈ 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 |Z | and |Z0| 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 ∈ R .

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 9 / 25 The multivariate skew normal distribution

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) ∈ R . Variance- mean mixture of CSN It can be shown that X1, ··· , Xn are identically distributed as distributions α References Xi ∼ SN 0, 1, √ , i = 1, ··· , n. 1+(n−1)α2 Note that √ α → 0 as n → ∞. 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 ∈ , W ∼ GIG(λ, χ, ψ), 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 ∈ 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 ∈ R , Σ 2 is the square root of a positive deﬁnite References matrix Σ ∈ Mp×p, Z ∼ SNp(0, Ip, α) and W ∼ GIG(λ, χ, ψ).

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, ψ), µ, γ ∈ Rp, denoted by X ∼ VMMCSNp,m(θ), θ = (λ, χ, ψ, 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 ∈ R such that ψ − 2t γ − t Σt > 0, where MGIG(·) is the moment generating function of GIG(λ, χ, ψ) and F1(·) is the cumulative distribution function of the p-dimensional GH (λ, χ, ψ − 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 + ψ, 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. Zhu depends on its parameters θ = (λ, χ, ψ, µ, γ, Σ, D, ∆), we plot the density function f (x) given by Equation (4) for Introduction of skew normal various different parameters with p = 1, 2 and m = 1, 2, 4 distributions using the R package ghyp by Breymann and Lüthi Variance- mean mixture (Breymann and Lüthi, 2013). of CSN distributions

References

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 18 / 25 Example

Variance- mean mixture In the following ﬁgures, we compare density functions of X of CSN X. Zhu for different D’s with λ = χ = ψ = γ = Σ = 1, µ = 0 and ∆ = Im, m = 1, 2. Introduction of skew normal distributions

Variance- 0.20 D=1' D=−1' 0.30 mean mixture D=(1, 1)' D=(−1, −1)' of CSN 0.25 distributions 0.15 0.20 References f(x) f(x) 0.10 0.15 0.10 0.05 0.05 0.00 0.00

0 5 10 15 0 5 10 15

x x

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 19 / 25 Example

Variance- mean mixture of CSN X. Zhu D=−5' D=5' D=(−5, 5)' D=(−5, −5)' 0.30 Introduction of 0.30 skew normal

distributions 0.20 0.20 f(x) f(x) Variance- 0.10 mean mixture 0.10 of CSN distributions 0.00 0.00

References 0 5 10 15 0 5 10 15

x x

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 20 / 25 Example

Variance- mean mixture In the following ﬁgure, we provide density function when of CSN X. Zhu m = 2 with the following parameters except indicated in 0 ﬁgures, λ = χ = ψ = γ = Σ = 1, µ = 0, D = (1, 1) , ∆ = I2, Introduction of skew normal ∆1 = I2, distributions Variance- 3, 2 5, −2 mean mixture ∆ = and ∆ = . of CSN 2 1, 2 3 3, 5 distributions

References

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 21 / 25 Example

Variance- mean mixture of CSN X. Zhu

0.20 γ=1 D=(5, 3)' 0.20 γ=3 D=(−3, −4)' Introduction of γ=−3 D=(2, −2)' 0.15 skew normal 0.15 distributions f(x) f(x) 0.10 0.10 Variance- mean mixture 0.05 of CSN 0.05 distributions 0.00 0.00

References −20 −10 0 10 20 0 5 10 15

x x

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 22 / 25 Example

Variance- mean mixture of CSN X. Zhu

0.20 ∆1 D=(5, 3)', ∆1 0.20

∆2 D=(−3, −4)', ∆2 ∆ ∆ Introduction of 3 D=(−3, −4)', 3 0.15 skew normal 0.15 distributions f(x) f(x) 0.10 0.10 Variance- mean mixture 0.05 of CSN 0.05 distributions 0.00 0.00

References 0 5 10 15 0 5 10 15

x x

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 23 / 25 Variance-mean mixture of CSN distributions

Variance- mean mixture of CSN Proposition X. Zhu Let X ∼ VMMCSNp,m(θ). For any A ∈ Mk×p of rank k and k Introduction of b ∈ , skew normal R distributions 0 Variance- AX + b ∼ VMMCSNk,m(λ, χ, ψ, Aµ + b, Aγ, AΣA , DA, ∆A), mean mixture of CSN distributions 0 −1 0 0 −1 0 where DA = DΣA ΣA , ∆A = ∆ + DΣD − DΣA ΣA AΣD . References

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 24 / 25 Variance- mean mixture of CSN X. Zhu

Introduction of skew normal distributions

Variance- mean mixture of CSN distributions Thank you! References

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 25 / 25 Variance- ——————————– mean mixture of CSN O. Arslan. Variance-mean mixture of the multivariate skew X. Zhu normal distribution. Statistical Papers, 56(2):353–378, Introduction of 2015. skew normal distributions A. Azzalini. A class of distributions which includes the Variance- mean mixture normal ones. Scandinavian Journal of Statistics, pages of CSN distributions 171–178, 1985. References A. Azzalini and A. Capitanio. Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(3):579–602, 1999. A. Azzalini and A. Dalla Valle. The multivariate skew-normal distribution. Biometrika, 83(4):715–726, 1996. W. Breymann and D. Lüthi. ghyp: A package on generalized X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 25 / 25 Variance- hyperbolic distributions. Manual for R Package ghyp, mean mixture of CSN 2013. X. Zhu G. González-Farías, A. Domínguez-Molina, and A. K.

Introduction of Gupta. Additive properties of skew normal random skew normal distributions vectors. Journal of Statistical Planning and Inference,

Variance- 126(2):521–534, 2004. mean mixture of CSN F. Vilca, N. Balakrishnan, and C. B. Zeller. Multivariate distributions skew-normal generalized hyperbolic distribution and its References properties. Journal of Multivariate Analysis, 128:73–85, 2014.

X. Zhu (UNA) Variance-mean mixture of CSN October 10, 2020 25 / 25