Inferences on location parameters based on independent multivariate skew normal distributions

Abstract

Purpose

The purpose of this study is to extend the classical noncentral F-distribution under normal settings to noncentral closed skew F-distribution for dealing with independent samples from multivariate skew normal (SN) distributions.

Design/methodology/approach

Based on generalized Hotelling's T² statistics, confidence regions are constructed for the difference between location parameters in two independent multivariate SN distributions. Simulation studies show that the confidence regions based on the closed SN model outperform the classical multivariate normal model if the vectors of skewness parameters are not zero. A real data analysis is given for illustrating the effectiveness of our proposed methods.

Findings

This study’s approach is the first one in literature for the inferences in difference of location parameters under multivariate SN settings. Real data analysis shows the preference of this new approach than the classical method.

Research limitations/implications

For the real data applications, the authors need to remove outliers first before applying this approach.

Practical implications

This study’s approach may apply many multivariate skewed data using SN fittings instead of classical normal fittings.

Originality/value

This paper is the research paper and the authors’ new approach has many applications for analyzing the multivariate skewed data.

Citation

Ma, Z., Wang, T., Wei, Z. and Zhu, X. (2022), "Inferences on location parameters based on independent multivariate skew normal distributions", Asian Journal of Economics and Banking, Vol. 6 No. 2, pp. 270-281. https://doi.org/10.1108/AJEB-03-2022-0034

1. Introduction

Although the normal distribution is a standard assumption for modeling observations in general, practitioners and researchers prefer more flexible models that account for the non-normality when the data collected in finance and econometric fields. The family of skew normal (SN) distributions, introduced by Azzalini (1985) for the univariate case, Azzalini and Valle (1996) for the multivariate case and Chen and Gupta (2005) for the matrix variate case, becomes a popular parametric family in statistical analysis of real data which account for asymmetry. There are several successful applications using SN, like modeling skewness premium of a financial asset by Carmichael and Coën (2013), addressing “wrong skewness” problems in stochastic frontier models by Wei et al. (2021). Here just list a few, an updated review was given by Adcock and Azzalini (2020).

Based on the definition given in Arellano-Valle et al. (2005), a p-dimensional random vector Y is said to be SN distributed with the location parameter vector μ∈Rp, the scale parameter matrix Σ (a p × p positive definite matrix), and the shape parameter vector λ∈Rp, denoted as Y∼SNpμ,Σ,λ, if its probability density function (pdf) is given by

(1)fYy=2ϕpy;μ,ΣΦλ′Σ−1/2y−μ,y∈Rp,

where ϕ_p(∙; μ, Σ) is the pdf of the p-dimensional normal distribution with the mean vector μ and the covariance matrix Σ, and Φ(∙) is the cumulative distribution function (cdf) of the univariate standard normal distribution. Extensions of Equation (1) are investigated by many researchers (see Azzalini and Capitanio, 1999; Wang et al., 2009; Young et al., 2016; Li et al., 2018).

For the univariate SN family, constructing plausibility regions for skewness parameter was discussed by Zhu et al. (2017) using inferential models (IMs). The joint plausibility regions for location parameter and skewness parameter were studied by Ma et al. (2018) when scale parameter is known using IMs, and the joint plausibility regions for location parameter and scale parameter were constructed by Zhu et al. (2018) when skewness parameter is given. For multivariate SN model, the confidence regions for location parameter are obtained by Ma et al. (2019). In this work, we study the difference of location parameters based on independent multivariate SN distributions so that the generalized Hotelling's T², and noncentral closed skew F-distributions are used. Under the assumption of equal but unknown scale parameters, the confidence regions for differences of location parameters of the multivariate SN model are proposed. Simulation studies show that the proposed confidence regions have higher relative coverage frequency rates than those in classical normal model for skewed data.

The organization of this paper is listed below. In Section 2, the definition of matrix variate SN distribution is introduced and some useful properties of sampling distribution on difference of sample means are derived. In Section 3, the confidence regions on the difference of location parameters by pivotal method are proposed when scale parameters from two populations are assumed to be equal but unknown. A group of simulation studies, which illustrate the effectiveness of our proposed methods, are given in Section 4, followed by a real data example in Section 5. The conclusion is given in Section 6.

2. Matrix variate SN distributions and sampling distributions

Let M_n×k be the set of all n × k matrices over the real field R and Rn=Mn×1. The transpose of a matrix A is denoted as A′. The n × n identity matrix is denoted as I_n, the constant vector (1,…,1)′∈Rn is denoted as 1_n, and J¯n=1n1n1n′. For B=(b1,b2,…,bn)′∈Mn×k with bi∈Rk, let Vec(B)=(b1′,b2′,…,bn′)′∈Rnk. For positive definite matrix T ∈ M_n×n, we use T⁻¹, T^1/2 and T^−1/2 to denote, respectively, the inverse, symmetric square root of T, and symmetric square root of T⁻¹. For B ∈ M_m×n, C ∈ M_n×p, we use B ⊗ C to denote the Kronecker product of B and C. Through this paper, N(0, 1) represents the standard normal distribution and bold phase letters represent vectors.

Definition 2.1.

Ye et al. (2014) The n × p random matrix Y is said to have a SN matrix distribution with location matrix M, scale matrix V ⊗Σ and skewness parameter matrix γ ⊗λ′, denoted by Y∼SNn×pM,V⊗Σ,γ⊗λ′, if y≡VecY∼SNnpμ,V⊗Σ,γ⊗λ, where M ∈ M_n×p, V ∈ M_n×n, μ=VecM, γ∈Rn and λ∈Rp

Suppose that X1∈Mn1×p and X2∈Mn2×p are two independent sample matrices such that

(2)Xi∼SNni×p1ni⊗μi′,Ini⊗Σi,1ni⊗λi′

for i = 1, 2. We are interested in analyzing the difference vector μ_d = μ₁ − μ₂. The sampling distributions of the sample mean and sample covariance matrix are given by Ma et al. (2019) in following Lemma.

Lemma 2.1.

Ma et al. (2019) Let Y∼SNn×p1n⊗μ′,In⊗Σ,1n⊗λ′. Then,

Y¯=1n1n′Y′∼SNpμ,Σn,nλ

and

(n−1)S=Y′In−J¯nY∼Wpn−1,Σ

are independently distributed, where Wpn−1,Σ represents the p-dimensional Wishart distribution with n − 1 degrees of freedom and the mean Σ.

By Lemma 2.1, we have

(3)X¯i=1ni1ni′Xi′∼SNpμi,Σini,niλi

and

(4)(ni−1)Si=Xi′Ini−J¯niXi∼Wpni−1,Σi

for i = 1, 2. It is natural to use the statistic X¯d=X¯1−X¯2 to inference on μ_d.

The difference between two independent SN distributed random vectors follows a closed SN distribution, which is reviewed below.

Definition 2.2.

(González-Farías et al. (Gonzalez-Farias et al., 2004) A random vector Y∈Rp is said to have closed SN distribution (CSN), denoted as CSNp,qμ,Σ,D,v,Δ, if its pdf is

fp,qy;μ,Σ,D,v,Δ=Cϕpy;μ,ΣΦqDy−μ;v,Δ,y∈Rp,

where C−1=Φq0;v,Δ+DΣD′, p ≥ 1, q ≥ 1, μ∈Rp, Σ∈Mp×p+, D ∈ M_q×p, v∈Rq, Δ∈Mq×q+ and ϕk⋅;η,Ω, Φk⋅;η,Ω are the pdf and cdf of a k-dimensional normal distribution.

For simplicity, we assume that ν = 0 so that Y ∼ CSN_p,q(μ, Σ, D, Δ). The following two properties of CSN can lead to the distribution of X¯d.

Lemma 2.2.

(González-Farías et al. (Gonzalez-Farias et al., 2004) Let Y ∼ CSN_p,q(μ, Σ, D, Δ)

1. For an arbitrary constant b∈Rp,

(5) Y+b∼CSNp,q(μ+b,Σ,D,Δ)

1. For nonzero real number c∈R,

(6) cY∼CSNp,q(cμ,c2Σ,c−1D,Δ);

1. Let Yi∼CSNp,qi(μi,Σi,Di,Δi), for i = 1, 2, be independently distributed. Then,

(7) Y1+Y2∼CSNp,q1+q2(μ1+μ2,Σ1+Σ2,D*,Δ*)

where

D*=D1Σ1Σ1+Σ2−1D2Σ2Σ1+Σ2−1,Δ*=A11A12A21A22

and

A11=Δ1+D1Σ1D1′−D1Σ1Σ1+Σ2−1Σ1D1′,A22=Δ2+D2Σ2D2′−D2Σ2Σ1+Σ2−1Σ2D2′,A12=−D1Σ1Σ1+Σ2−1Σ2D2′.

In term of CSN, X¯i∼CSNp,1μi,Σini,niλi′Σi−1/2,1 for i = 1, 2. Thus, we obtain the distribution of X¯d.

Theorem 2.1.

Let X¯d=X¯1−X¯2 with X¯i∼CSNp,1μi,Σini,niλi′Σi−1/2,1 for i = 1, 2. Then,

(8) X¯d∼CSNp,2μd,Σd,Dd,Δd

where

(9) μd=μ1−μ2,Σd=Σ1n1+Σ2n2Dd=λ1′Σ11/2Σd−1−λ2′Σ21/2Σd−1,Δd=A11A12A21A22

with

A11=1+n1λ1′λ1−λ1′Σ11/2Σd−1Σ11/2λ1,A22=1+n2λ2′λ2−λ2′Σ21/2Σd−1Σ21/2λ2,A12=A21=λ1′Σ11/2Σd−1Σ21/2λ2.

Proof.

By part (2) and (3) of Lemma 2.2, the desired result follows immediately. □

Remark.

If λ₂ = 0, i.e. X₂ following multivariate normal distribution with mean μ₂ and covariance Σ2n2, the distribution of difference X¯d has the form Xd∼CSNp,1(μd,Σd,λ1′Σ11/2Σd1/2,A11) which can be further expressed as Xd∼SNp(μd,Σd,Σ11/2λ1/A111/2).

Figure 1 presents the contour of bivariate closed SN for various combinations of shape parameter parameters D with different scale parameter Σ=1ρρ1. Specifically, the gray contours show D = 0 with ρ = 0, 0.5 and −0.5; the green contours show D=0030 with ρ = 0, 0.5 and −0.5; the blue contours show D=1−500 with ρ = 0, 0.5 and −0.5 the red contours show D=1−530 with ρ = 0, 0.5 and −0.5. From another point of view, these contour plots present bivariate normal distribution (gray), SN distribution (green and blue) and closed SN distribution (red).

3. Inference on difference of location parameters

In this section, the inference on the difference of location parameter is proposed when the scale parameter Σ₁ and Σ₂ are unknown but assumed to be equal, say Σ₁ = Σ₂ = Σ. The main result is based on the generalized Hotelling's T² under multivariate SN setting.

3.1 Some related distributions

At first, we consider the distribution of S=X¯d−μd′Σd−1X¯d−μd. The following definition and lemma by Zhu et al. (2019) are useful to derive the distribution of S.

Definition 3.1.

Zhu et al. (2019) Let X ∼ CSN_p,q(μ, I_p, D, Δ_q). The distribution of X′X, denoted by X′X∼CSχp2(λ,δ1,δ2,Δq), is called a noncentral closed skew chi-square distribution with degrees of freedom p, noncentrality parameter λ = μ′μ, skewness parameters δ₁ = Dμ, δ₂ = DD′ and parameter Δ_q

Lemma 3.1.

Zhu et al. (2019) Let X ∼ CSN_p,q(μ, Σ, D, Δ) and Q = X′WX with a nonnegative definite W ∈ M_p×p. If Σ^1/2WΣ^1/2 is idempotent of rank k, then Q∼CSχk2(λ,δ1,δ2,Ωq), where λ = μ′Wμ, δ₁ = DΣWμ, δ₂ = DΣWΣD′ and Ω_q = Δ + D(Σ −ΣWΣ)⁻¹D′

Based on Theorem 2.1 and Lemma 3.1, we obtain the following result.

Proposition 3.1.

Let S=X¯d−μd′Σd−1X¯d−μd. Then, S∼CSχp2(0,0,δ2,Ω) with

δ2=n1n2n1+n2λ1′λ1λ2′λ1λ2′λ1λ2′λ2 and Ω=1+n12λ1′λ1n1+n2n1n2λ2′λ1n1+n2n1n2λ2′λ1n1+n21+n22λ2′λ2n1+n2.

Proof.

From part (i) of Lemma 2.2, we have X¯d−μd∼CSNp,2(0,Σd,Dd,Δd). Since Σd1/2Σd−1Σd1/2=Ip is an idempotent of rank p, the desired result follows immediately. □

Remark.

Comparing with one sample case, the distribution of quantity n(X¯−μ)′Σ−1(X¯−μ)∼χp2 for X¯∼SNpμ,Σn,nλ is free of the skewness parameter λ. Here, the distribution of S follows noncentral closed SN distribution given above which depends on the parameters δ₂ and Ω. Readers are referred to check out Figures 5 and 6 in Zhu et al. (2019) for the density curves of CSχ²(0, 0, δ₂, Ω).

3.2 Confidence region of μ_d

In this subsection, we will extend the Hotelling's T² statistic from multivariate normal setting to the multivariate SN setting, called the generalized Hotelling's T2=n1n2n1+n2(X¯d−μd)′Sp−1(X¯d−μd), to construct the confidence regions for the difference of location vector where (n1+n2−2)Sp=n1−1S1+n2−1S2. First we need to derive the distribution of S_p, then extend the F-distribution to closed skew F-distribution which can describe the distribution of T² under the multivariate SN setting.

Proposition 3.2.

Let (n1+n2−2)Sp=n1−1S1+n2−1S2 with S₁ and S₂ defined by equation (4) Then, (n₁ + n₂ − 2)S_p ∼ W_p(n₁ + n₂ − 2, Σ)

Proof.

By Lemma 2.1, (n_i − 1)S_i ∼ W_p(n_i − 1, Σ) for i = 1, 2 are independently distributed. Thus, the well-known properties of Wishart distribution for sums and scale transformation lead to the desired result. □

To obtain the distribution of T², we need the following well-known result (Lemma 3.2, Mardia et al. (1980), Theorem 3.4.7) and extended version of the F-distribution, called closed skew F-distribution, Definition 3.2, which was introduced by Zhu et al. (2019).

Lemma 3.2.

If H∼Wpm,Σ, m > p, then the ratio a′Σ−1a/a′H−1a∼χm−p+12 for any fixed p-vector a

Definition 3.2.

Zhu et al. (2019) Let U1∼CSχn12(λ,δ1,δ2,Δm), U2∼χn22, and U₁ and U₂ are independent. The distribution of F=U1/n1U2/n2 is called the noncentral closed skew F-distribution with degrees of freedom n₁ and n₂, and parameters λ, δ₁, δ₂ and Δ_m, denoted by F∼CSFn1,n2(λ,δ1,δ2,Δm)

Based on above definition, the pdf of noncentral closed skew F-distribution can be obtained below.

Proposition 3.3.

Let F∼CSFn1,n2(λ,δ1,δ2,Δm). The pdf of F is given by

(10) fF(x;λ,δ1,δ2,Δm)=∫0∞n1vn2f1n1xvn2;λ,δ1,δ2,Δmf2(v)dv

where f₁(∙; λ, δ₁, δ₂, Δ_m) and f₂(∙) are pdf of CSχn12(λ,δ1,δ2,Δm) and χn22, respectively.

Proof.

Let F=U1/n1U2/n2 with U1∼CSχn12(λ,δ1,δ2,Δm), U2∼χn22 independently distributed. The joint density of (U₁, U₂) is f(U1,U2)(u,v)=f1(u)f2(v) where f₁(∙) and f₂(∙) are pdf of U₁ and U₂, respectively. Then change of variables x=n2un1v, h = v, for x > 0, h > 0. The Jacobian of this transformation is n₁v/n₂. So integrating with respect to v over 0 < v < ∞ leads to desired results.

By Lemma 3.2 and Definition 3.2, we obtain the distribution of T² as follows.

Theorem 3.1.

For two independently distributed random matrices

Xi∼SNni×p1ni⊗μi′,Ini⊗Σ,1ni⊗λi′fori=1,2,

where λ_i's are known. Let X¯i, S_i be given by equation (3) and (4), X¯d=X¯1−X¯2, and Sp=n1−1S1+n2−1S2. Then, the distribution of T2=n1n2n1+n2(X¯d−μd)′Sp−1(X¯d−μd) is given by

(11) n1+n2−p−1p(n1+n2−2)T2∼CSFp,n1+n2−p−1(0,0,δ2,Ω),

where δ2=n1n2n1+n2λ1′λ1λ2′λ1λ2′λ1λ2′λ2 and Ω=1+n12λ1′λ1n1+n2n1n2λ2′λ1n1+n2n1n2λ2′λ1n1+n21+n22λ2′λ2n1+n2.

Proof.

Rewrite T² as

T2=n1n2n1+n2X¯d−μd′Σ−1X¯d−μdX¯d−μd′Σ−1X¯d−μd/X¯d−μd′Sp−1X¯d−μd.

Since S_p and X¯d−μd are independent by Lemma 2.1, the conditional distribution of

w=(n1+n2−2)X¯d−μd′Σ−1X¯d−μd/X¯d−μd′Sp−1X¯d−μd∼χn1+n2−p−12

given X¯d−μd by Proposition 3.2 and Lemma 3.2. It is clear that w∼χn1+n2−p−12 since the conditional distribution does not depend on X¯d−μd. On the other hand, since Σd−1=n1n2n1+n2Σ−1, then n1n2n1+n2X¯d−μd′Σ−1X¯d−μd=X¯d−μd′Σd−1X¯d−μd which follows closed skew chi-square distribution CSχp2(0,0,δ2,Ω) from Proposition 3.1. Therefore, the desired result follows by Definition 3.2.

Based on above results, we construct confidence regions for the difference of location parameter μ_d by using generalized Hotelling's T² as a pivotal statistics.

Theorem 3.2.

Assume two samples satisfying (2) with unknown Σ₁ and Σ₂ but Σ₁ = Σ₂ and known λ₁ and λ₂. Then, the 1001−α% confidence regions for μ_d is given by

CμdPα=μd:n1+n2−p−1p(n1+n2−2)T2<CSFp,n1+n2−p−121−α,

where CSFp,n1+n2−p−121−α represents the 1 − α quantile of CSFp,n1+n2−p−12(0,0,δ2,Ω)

The following plots present the pdf of noncentral closed skew F-distribution (see Figure 2).

4. Simulation study

Simulations are conducted for evaluating the performance of the proposed confidence regions for μ_d under independent multivariate SN settings using the coverage relative frequency rates. Comparisons of proposed confidence regions with those in classical independent multivariate normal distributions are given.

4.1 Coverage frequencies

To evaluate the proposed confidence regions for difference of location parameters under multivariate SN setting, Monte Carlo simulation studies (each with a number of simulation runs M = 10,000) are conducted for combinations of various values of sample sizes (n₁, n₂) = (20,25), (40, 50) and (80,100), (ρ₁, ρ₂) = ({0.1, 0.5, 0.8}, {0.1, 0.5, 0.8}), D1=132−5 and D2=−2230.

Table 1 shows that our method provides reliable inference about the difference of location parameters with nominal confidence level (95%). To further illustrate the effectiveness of the proposed method, we confidence intervals of the coverage probability presented in the following plot.

From Figure 3, we can see clearly that the pivotal quantity-based closed skew F-distribution produce more robust confidence region than that based on F-distribution. The coverage relative frequencies, based on the SN model, are close to the nominal confidence level 95% consistently for the combination of different sample sizes, scale parameter and skewness parameters. But the coverage relative frequencies, based on the normal model, are lower than the nominal confidence level.

5. Real data example

In this section, we illustrate the effectiveness and applicability of the proposed methods by applying them to Australian Institute of Sport (AIS) data (Cook and Weisberg, 2009). We explore the difference of body mass index (BMI) and lean body mass (LMB) between males and females athletes in AIS data.

The point estimates of the parameters for AIS data are reported in Table 2.

In Figure 4, the scatter plots and contour plots of fitted bivariate SN distributions are presented. Based on our previous work (Azzalini and Valle, 1996), this data set prefers multivariate SN model. So we adopt multivariate SN model as well to explore the difference of location parameters. Using point estimates listed in Table 2 and applying Theorem 2.1, the differences of sample mean has closed SN distribution, X¯d∼CSN2,2(μ,Σ,D,Δ) with estimated parameters

μˆ=(1.29,18.13)′,Σˆ=0.130.410.412.09

Dˆ=3.20−4.83−29.8016.82,Δˆ=28.90101.01101.01473.42.

Then, we apply Theorem 3.2 to construct the confidence region of difference of location parameters μ_d. In Figure 5, the confidence regions for the difference of location parameters are given below at 95% confidence level.

6. Conclusion

In this work, the difference for location parameters between two independent samples under multivariate SN setting is studied. The construction of confidence region procedure is developed. From the results of simulation studies, the confidence region based on SN model has better performance than normal model in term of relative coverage frequencies to capture the true value when the data are generated from skewed distribution.

Figures

Figure 1

Contour plot of bivariate normal, SN and close SN distributions with various values of parameters

Figure 2

The pdfs of noncentral closed skew F2,n(5,(5,11)T,5111125,10.50.51) distributions with n = 5, 10, 20 and 50 (black, green, red and blue)

Figure 3

The confidence intervals of coverage relative frequencies at confidence level α = 0.95, red ones based on SN model and blue ones based on normal model with sample size (n₁, n₂) = (20, 25), (40, 50), (80, 100) and (200, 250) (from left to right in each figure), ρ = 0.2, 0.5 and 0.8 (in each row), and D₁ and D₂ (from left to right), respectively

Figure 4

The scatter plots and contour plots for the AIS data of males and females

Figure 5

The confidence region of difference of location parameter μ_d at confidence level 95% for AIS data between males and females, where the black triangle represents the μˆd , and the blue dot represents the μd0

Table 1

Coverage relative frequencies of confidence regions at confidence level α = 95% for difference of location parameters μ_d with various combinations of sample sizes, ρ₁, ρ₂ and D₁, D₂ using Hotelling's T² as the pivotal quantity when Σ₁ and Σ₂ are equal but unknown

n1,n2	Σ=1ρρ1	D1=132−5		D2=−2230
		F-distribution	CSF-distribution	F-distribution	CSF-distribution
20,25	ρ = 0.2	0.9268	0.9495	0.9224	0.9524
	ρ = 0.5	0.9274	0.9511	0.9135	0.9502
	ρ = 0.8	0.9206	0.9504	0.9227	0.9520
40,45	ρ = 0.2	0.9025	0.9479	0.9008	0.9534
	ρ = 0.5	0.9118	0.9451	0.9095	0.9509
	ρ = 0.8	0.9036	0.9521	0.9016	0.9524
80,100	ρ = 0.2	0.8932	0.9527	0.9080	0.9479
	ρ = 0.5	0.8917	0.9463	0.8702	0.9470
	ρ = 0.8	0.9035	0.9483	0.8933	0.9448
200,250	ρ = 0.2	0.8824	0.9503	0.8883	0.9461
	ρ = 0.5	0.88920	0.9533	0.8956	0.9559
	ρ = 0.8	0.8998	0.9492	0.8852	0.9546

Table 2

Point estimates of SN parameters for the males and females AIS data, respectively

	Males	Females
μˆ′	24.47,79.55	23.18,61.42
Σˆ	4.8118.9518.95114.00	8.3221.3021.3089.96
λˆ′	−0.20,−0.80	0.88,−2.63