Inferences on location parameters based on independent multivariate skew normal distributions

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

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.

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.

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

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.

By Lemma 2.1, we have

and

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.

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 $X¯d$⁠.

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

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).

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.

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.

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

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.

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).

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

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

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

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.

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

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.

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.

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

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.

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.

The authors would like to thank reviewers for their valuable suggestions and comments, which improved the manuscript.