Addressing non-normality in multivariate analysis using the t-distribution

Abstract

The main aim of this paper is to propose a set of tools for assessing non-normality taking into consideration the class of multivariate t-distributions. Assuming second moment existence, we consider a reparameterized version of the usual t distribution, so that the scale matrix coincides with covariance matrix of the distribution. We use the local influence procedure and the Kullback–Leibler divergence measure to propose quantitative methods to evaluate deviations from the normality assumption. In addition, the possible non-normality due to the presence of both skewness and heavy tails is also explored. Our findings based on two real datasets are complemented by a simulation study to evaluate the performance of the proposed methodology on finite samples.

Appendices

Appendix A: The expected information matrix

We use the results shown in the Supplementary Material and note that the Fisher information matrix for $\varvec{\theta }$ can be written as

$$\begin{aligned} \varvec{{\mathcal {I}}}(\varvec{\theta }) = \frac{1}{n}\sum _{i=1}^n {{\,\mathrm{E}\,}}\{\varvec{U}_i (\varvec{\theta }) \varvec{U}_i^\top (\varvec{\theta })\}, \end{aligned}$$

where the score function $\varvec{U}_i(\varvec{\theta }) =(\varvec{U}_i^\top (\varvec{\mu }),\varvec{U}_i^\top (\varvec{\phi }), U_i(\eta ))^\top$ associated to the ith component of the log-likelihood, with $i=1,\dots ,n$, is defined in Equations (4)-(6), and the expected value ${{\,\mathrm{E}\,}}(\cdot )$ is taken with respect to the density function in (1). Next, we obtain each of the blocks of the Fisher information matrix reported in Equation (7).

From the score functions (4) and (5), it follows that

$$\begin{aligned} {{\,\mathrm{E}\,}}\{\varvec{U}_i(\varvec{\mu })\varvec{U}_i^\top (\varvec{\mu })\}&= c_\mu (\eta )\varvec{\Sigma }^{-1},\\ {{\,\mathrm{E}\,}}\{\varvec{U}_i(\varvec{\phi })\varvec{U}_i^\top (\varvec{\phi })\}&= \frac{1}{4}\varvec{D}_p^\top \left\{ 2c_\phi (\eta ) (\varvec{\Sigma }^{-1}\otimes \varvec{\Sigma }^{-1})\varvec{N}_p\right. \\&\quad \left. + (c_\phi (\eta ) - 1)({{\,\mathrm{vec}\,}}\varvec{\Sigma }^{-1}) ({{\,\mathrm{vec}\,}}\varvec{\Sigma }^{-1})^\top \right\} \varvec{D}_p, \end{aligned}$$

where $c_\mu (\eta )=c_\phi (\eta )/(1-2\eta )$, $c_\phi (\eta )=(1+p\eta )/(1+(p+2)\eta )$. Note that $c_\mu (\eta )$ and $c_\phi (\eta )\rightarrow 1$ when $\eta \rightarrow 0$. On the other hand, we have that $\varvec{N}_p\varvec{D}_p=\varvec{D}_p$ (see Magnus and Neudecker 1999), which produces the expressions corresponding to the normal case.

Therefore, it is clear that

$$\begin{aligned} \frac{\partial v_i}{\partial \eta } = (1-2\eta )^{-3}\left\{ (p+2) (1-2\eta )q_i^{-1} - (1 + \eta p)q_i^{-2}\delta _i^2\right\} , \end{aligned}$$

with $q_i = 1+c(\eta )\delta _i^2$. Thus, we obtain

$$\begin{aligned} \frac{\partial \varvec{U}_i(\varvec{\mu })}{\partial \eta }&= (1-2\eta )^{-3}\varvec{\Sigma }^{-1} \left\{ (p+2)(1 - 2\eta ) q_i^{-1}\varvec{Z}_i - (1 + \eta p)q_i^{-2}\delta _i^2\varvec{Z}_i\right\} , \\ \frac{\partial \varvec{U}_i(\varvec{\phi })}{\partial \eta }&=\frac{1}{2}(1 - 2\eta )^{-3}\varvec{D}_p^\top {{\,\mathrm{vec}\,}}\left\{ \varvec{\Sigma }^{-1}\left( (p+2)(1 - 2\eta )q_i^{-1}\varvec{Z}_i\varvec{Z}_i^{\top }\right. \right. \\&\quad \left. \left. - (1 + \eta p)q_i^{-2}\delta _i^2\varvec{Z}_i\varvec{Z}_i^{\top }\right) \varvec{\Sigma }^{-1}\right\} . \end{aligned}$$

By applying Lemmas 3 and 5 from Supplementary Material, it follows that

$$\begin{aligned} {{\,\mathrm{E}\,}}\left\{ \frac{\partial \varvec{U}_i(\varvec{\mu })}{\partial \eta }\right\}&= \varvec{0}, \\ {{\,\mathrm{E}\,}}\left\{ \frac{\partial \varvec{U}_i(\varvec{\phi })}{\partial \eta }\right\}&= \frac{c(\eta )(p+2)}{(1 + \eta p)(1 + (p+2)\eta )}\varvec{D}_p^\top {{\,\mathrm{vec}\,}}(\varvec{\Sigma }^{-1}), \end{aligned}$$

for $i=1,\dots ,n$. The score function for $\eta$ can be written as,

$$\begin{aligned} U_i(\eta )&= \frac{1}{2\eta ^2}\left\{ pc(\eta ) + \psi \left( \frac{1}{2\eta }\right) -\psi \left( \frac{1+p\eta }{2\eta }\right) - \frac{1+p\eta }{1-2\eta } \frac{c(\eta )\delta _i^2}{1 + c(\eta )\delta _i^2} +\log (1+c(\eta )\delta _i^2)\right\} , \\&= \frac{1}{2\eta ^2}\left\{ \log (1+ Q_{i\eta }) -\left( \psi \left( \frac{1+p\eta }{2\eta }\right) - \psi \left( \frac{1}{2\eta }\right) \right) -\left( \frac{1+p\eta }{1-2\eta }\frac{Q_{i\eta }}{1 + Q_{i\eta }} - pc(\eta )\right) \right\} , \end{aligned}$$

where $Q_{i\eta } = c(\eta ) \delta _i^2\sim \chi _p^2/\chi _{1/\eta }^2$, ${{\,\mathrm{E}\,}}\{Q_{i\eta } (1+Q_{i\eta })^{-1}\}=\frac{p\eta }{1+p\eta }$ and ${{\,\mathrm{E}\,}}\{\log (1 + Q_{i\eta })\}=\psi \left( \frac{1+p\eta }{2\eta }\right) -\psi \left( \frac{1}{2\eta }\right)$. Let $U_1 =\log (1+Q_{i\eta })$, $U_2 =\frac{1+p\eta }{1 - 2\eta }\frac{Q_{i\eta }}{1+Q_{i\eta }}$, $\overline{U}_1 = U_1 - {{\,\mathrm{E}\,}}(U_1)$ and $\overline{U}_2 = U_2 -{{\,\mathrm{E}\,}}(U_2)$. Then,

$$\begin{aligned} {{\,\mathrm{E}\,}}\{U_i(\eta )\}&= {{\,\mathrm{E}\,}}\{(U_1 - {{\,\mathrm{E}\,}}(U_1)) - (U_2 - {{\,\mathrm{E}\,}}(U_2))\} = 0,\\ {{\,\mathrm{var}\,}}\{U_i(\eta )\}&= \frac{1}{4\eta ^4}\{{{\,\mathrm{E}\,}}(\overline{U}_1^2) -2{{\,\mathrm{E}\,}}(\overline{U}_1 \overline{U}_2) + {{\,\mathrm{E}\,}}(\overline{U}_2^2)\} \\&= \frac{1}{4\eta ^4}\{{{\,\mathrm{var}\,}}(\overline{U}_1) -2{{\,\mathrm{Cov}\,}}(\overline{U}_1,\overline{U}_2) + {{\,\mathrm{var}\,}}(\bar{U}_2)\} \\&= \frac{1}{4\eta ^4}\{{{\,\mathrm{E}\,}}(U_1^2) - {{\,\mathrm{E}\,}}^2(U_1) - 2({{\,\mathrm{E}\,}}(U_1 U_2) -{{\,\mathrm{E}\,}}(U_1){{\,\mathrm{E}\,}}(U_2)) + {{\,\mathrm{E}\,}}(U_2^2)-{{\,\mathrm{E}\,}}^2(U_2)\}. \end{aligned}$$

Using the fact that $\psi (x+1) = \psi (x)+1/x$ we have

$$\begin{aligned} {{\,\mathrm{E}\,}}(U_1^2)&= {{\,\mathrm{E}\,}}\{(\log (1+Q_{i\eta }))^2\} = {{\,\mathrm{E}\,}}^2(U_1) - \psi '\left( \frac{1+p\eta }{2\eta }\right) + \psi '\left( \frac{1}{2\eta }\right) , \\ {{\,\mathrm{E}\,}}\{U_1 U_2)&= \frac{1+p\eta }{1-2\eta }{{\,\mathrm{E}\,}}\{Q_{i\eta } (1 + Q_{i\eta })^{-1}\log (1 + Q_{i\eta })\} \\&= \left\{ {{\,\mathrm{E}\,}}(U_1) + \frac{2\eta }{1+p\eta }\right\} {{\,\mathrm{E}\,}}(U_2), \\ {{\,\mathrm{E}\,}}(U_2^2)&= \left( \frac{1+p\eta }{1-2\eta }\right) ^2 {{\,\mathrm{E}\,}}\{Q_{i\eta }^2(1+Q_{i\eta })^{-2}\} \\&= \frac{p+2}{p}\frac{1+p\eta }{1+(p+2)\eta }{{\,\mathrm{E}\,}}(U_2)^2. \end{aligned}$$

Finally, we have that the expected information in relation to $\eta$ is given by,

$$\begin{aligned} {{\,\mathrm{var}\,}}\{U_i(\eta )\}&= \frac{1}{4\eta ^4}\left\{ -\psi '\left( \frac{1+p\eta }{2\eta }\right) + \psi '\left( \dfrac{1}{2\eta }\right) - \frac{4\eta }{1+p\eta }{{\,\mathrm{E}\,}}(U_2) \right. \\&\quad \left. + \frac{p+2}{p}\frac{1+p\eta }{1+(p+2)\eta }{{\,\mathrm{E}\,}}(U_2)^2 - {{\,\mathrm{E}\,}}(U_2)^2\right\} \\&= \frac{1}{4\eta ^4}\left\{ \psi '\left( \frac{1}{2\eta }\right) - \psi ' \left( \frac{1+p\eta }{2\eta }\right) + 2pc(\eta )^2\left( \frac{4(p+2) \eta ^2 - p\eta -1}{(1+p\eta )(1+(p+2)\eta )}\right) \right\} . \end{aligned}$$

From the expansion (Abramowitz and Stegun 1970, Sec. 6.4.12),

$$\begin{aligned} \psi '(x)&= \frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} + O\left( \frac{1}{x^5}\right) \quad \text {as} \quad x\rightarrow \infty , \\ (1+ax)^{-k}&= 1-k a x + \frac{k(k+1)}{2}a^2x^2 -\frac{k(k+1)(k+2)}{6}a^3x^3 + O(x^4) \quad \text {as} \quad x\rightarrow 0, \end{aligned}$$

we find as $\eta \rightarrow 0$ that,

$$\begin{aligned} \psi '\left( \frac{1}{2\eta }\right) - \psi '\left( \frac{1+p\eta }{2\eta }\right)&= 2\eta +2\eta ^2 + \frac{4}{3}\eta ^3 - 2\eta (1+p\eta )^{-1} - 2\eta ^2(1+p\eta )^{-2} \\&\quad - \frac{4}{3}\eta ^3(1+p\eta )^{-3} + O(\eta ^5) \\&= 2p\eta ^2 - (2p^2-4p)\eta ^3 + (2p^3-6p^2+4p)\eta ^4 + O(\eta ^5). \end{aligned}$$

Similarly,

$$\begin{aligned} 2pc(\eta )^2&\left( \frac{4(p+2)\eta ^2 - p\eta -1}{(1+p\eta )(1 + (p + 2)\eta )}\right) = \frac{2p\eta ^2(4(p+2)\eta ^2 - p\eta - 1)}{(1 - 2\eta )^2(1 + p\eta )(1 + (p+2)\eta )} \\&= (8p(p+2)\eta ^4 - 2p^2\eta ^3 - 2p\eta ^2)(1 + 4\eta + 12\eta ^2 + O(\eta ^3)) \\&\quad \times (1 - p\eta + p^2\eta ^2 + O(\eta ^3))(1 - (p+2)\eta - (p+2)^2\eta ^2 O(\eta ^3)) \\&= -2p\eta ^2 + (2p^2 - 4p)\eta ^3 + (2p^3+24p^2+16p)\eta ^4 + O(\eta ^5). \end{aligned}$$

Hence,

$$\begin{aligned} {{\,\mathrm{var}\,}}\{U_i(\eta )\} = \frac{1}{4}(4p^3+18p^2+20p) + O(\eta ) =\frac{p(p+2)(2p+5)}{2} + O(\eta ). \end{aligned}$$

Appendix B: Non-normality due to asymmetry

Another source of non-normality is the possible asymmetry present in the observations. Shannon entropy, Kullback–Leibler divergence and mutual information for multivariate skew-elliptical distributions have been considered in the literature, see for instance Arellano-Valle et al. (2012) and Contreras-Reyes and Arellano-Valle (2012). We summarize some of these results for the multivariate skew normal and skew t distributions below.

Following Arellano-Valle et al. (2012), we say that a random vector $\varvec{Z}\in {\mathbb {R}}^p$ has a skew-normal distribution with location vector $\varvec{\xi }\in {\mathbb {R}}^p$, dispersion matrix $\varvec{\Omega } > 0$ and shape/skewness parameter $\varvec{\gamma }\in {\mathbb {R}}^p$, denoted by $\varvec{Z}\sim \mathsf {SN}_p(\varvec{\xi },\varvec{\Omega },\varvec{\gamma })$, if its probability density function is

$$\begin{aligned} f(\varvec{z}) = 2\phi _p(\varvec{z};\varvec{\xi },\varvec{\Omega })\,\Phi \{\varvec{\gamma }^\top (\varvec{z}-\varvec{\xi })\}, \qquad \varvec{z} \in {\mathbb {R}}^p, \end{aligned}$$

(B.1)

where

$$\begin{aligned} \phi _p(\varvec{z};\varvec{\xi },\varvec{\Omega }) = (2\pi )^{-p/2}\vert \varvec{\Omega }\vert ^{-1/2}\exp (-\delta _{\mathsf{skew}}^2/2), \end{aligned}$$

is the probability density function of the p-variate $\mathsf {N}_p(\varvec{\xi },\varvec{\Omega })$ distribution, $\Phi (\cdot )$ is the univariate $\mathsf {N}(0,1)$ cumulative distribution function and $\delta _{\mathsf{skew}}^2 =(\varvec{z}-\varvec{\xi })^\top \varvec{\Omega }^{-1} (\varvec{z}-\varvec{\xi }) \sim \chi ^2(p)$. The vector of means and the covariance matrix of $\varvec{Z}$ are given, respectively, by

$$\begin{aligned} \varvec{\mu }_{\mathsf{SN}} = \varvec{\xi } + \sqrt{\frac{2}{\pi }}\varvec{\delta }, \qquad \text {and} \qquad \varvec{\Sigma }_{\mathsf{SN}} = \varvec{\Omega } - \frac{2}{\pi }\varvec{\delta }\varvec{\delta }^\top , \end{aligned}$$

where, $\varvec{\delta } = \varvec{\Omega \gamma }/\sqrt{1 + \tau ^2}$ and $\tau ^2=\varvec{\gamma }^\top \varvec{\Omega \gamma }$.

We say that a random vector $\varvec{Z}\in {\mathbb {R}}^p$ has a skew-t distribution with location vector $\varvec{\xi }\in {\mathbb {R}}^p$, dispersion matrix $\varvec{\Omega }\in {\mathbb {R}}^{p \times p}$, shape/skewness parameter $\varvec{\gamma }\in {\mathbb {R}}^p$ and $\nu > 0$ degrees of freedom, denoted by $\varvec{Z} \sim \mathsf {St}_p(\varvec{\xi },\varvec{\Omega },\varvec{\gamma },\nu )$, if its probability density function is given by

$$\begin{aligned} f(\varvec{z}) = 2t_p(\varvec{z};\varvec{\xi },\varvec{\Omega },\nu )\,T \left( \sqrt{\frac{\nu + p}{\nu + \delta _{\mathsf{skew}}^2}} \, \varvec{\gamma }^\top (\varvec{z} - \varvec{\xi }); \nu + p\right) , \end{aligned}$$

(B.2)

where

$$\begin{aligned} t_p(\varvec{z};\varvec{\xi },\varvec{\Omega },\nu ) = \frac{\Gamma \left( \frac{\nu + p}{2}\right) }{\Gamma \left( \frac{\nu }{2}\right) (\nu \pi )^{p/2}} \vert \varvec{\Omega }\vert ^{-1/2}\left( 1 + \frac{1}{\nu } \delta _{\mathsf{skew}}^2\right) ^{-(\nu +p)/2},\qquad \varvec{z}\in {\mathbb {R}}^p, \end{aligned}$$

is the probability density function of the p-variate $t_p(\varvec{\xi },\varvec{\Omega },\nu )$ distribution, $\delta _{\mathsf{skew}}^2 = (\varvec{z} - \varvec{\xi })^\top \varvec{\Omega }^{-1}(\varvec{z} - \varvec{\xi })/p \sim F(p,\nu )$ and $T(x;\nu + p)$ is the $T_1(0,1,\nu +p)$ cumulative distribution function (see, for instance Azzalini and Genton 2008; Arellano-Valle et al. 2012, for details).

If $\varvec{Z}\sim \mathsf {St}_p(\varvec{\xi },\varvec{\Omega },\varvec{\gamma },\nu )$ then the vector of means and the covariance matrix of $\varvec{Z}$ is given by

$$\begin{aligned} \varvec{\mu }_{\mathsf{St}}&= \varvec{\xi } + \alpha (\nu )\varvec{\delta }, \quad \nu> 1 \\ \varvec{\Sigma }_{\mathsf{St}}&= \frac{\nu }{\nu -2}\varvec{\Omega } - \{\alpha (\nu )\}^2 \varvec{\delta }\varvec{\delta }^\top , \quad \nu > 2, \end{aligned}$$

where $\alpha (\nu ) =\{\Gamma ((\nu -1)/2)/\Gamma (\nu /2)\} \sqrt{\nu /\pi }$. Note that $\alpha (\nu )\rightarrow \sqrt{2/\pi }$ as $\nu \rightarrow \infty$, and we obtain the results for the skew-normal distribution given above.

Arellano-Valle et al. (2012) show that for the skew-normal and skew-t distributions the Shannon entropy has explicit form and given in the following lemmas.

Lemma 4

If $\varvec{X}\sim \mathsf {SN}_p(\varvec{\xi },\varvec{\Omega },\varvec{\gamma })$ and $\varvec{Y}\sim \mathsf {St}_p(\varvec{\xi }, \varvec{\Omega },\varvec{\gamma },\nu )$, then the Shannon entropy is given by

• (i) $H(\varvec{X}) = \frac{1}{2}\log \vert \varvec{\Omega }\vert +\frac{p}{2}(1+ \log 2\pi ) - {{\,\mathrm{E}\,}}[\log \{2\Phi (\tau W)\}]$,

• (ii) $H(\varvec{Y}) = \frac{1}{2}\log \vert \varvec{\Omega }\vert -\log \Gamma \left( \frac{\nu + p}{2}\right) + \log \Gamma \left( \frac{\nu }{2}\right) + \frac{p}{2}\log (\nu \pi ) +\frac{\nu + p}{2}\left\{ \psi \left( \frac{\nu + p}{2}\right) -\psi \left( \frac{\nu }{2}\right) \right\} - {{\,\mathrm{E}\,}}[\log \{2T(\tau W^*; \nu +p)\}]$,

with $W \sim \mathsf {SN}(0,1,\tau )$, $\tau ^2 =\varvec{\gamma }^\top \varvec{\Omega \gamma }$; $W^* = \sqrt{\nu + p} \,W_{\mathsf{St}}/\sqrt{\nu + p-1 + W_{\mathsf{St}}^2}$ where $W_{\mathsf{St}} \sim \mathsf {St}(0,1,\tau ,\nu +p-1)$.

Lemma 5

Let $\varvec{Z}\sim \mathsf {N}_p(\varvec{\mu },\varvec{\Sigma })$. If $\varvec{X}\sim \mathsf {SN}_p(\varvec{\xi }, \varvec{\Omega },\varvec{\gamma })$, $\varvec{Y}\sim \mathsf {St}_p(\varvec{\xi },\varvec{\Omega },\varvec{\gamma },\nu )$ then the negentropy of $\varvec{X}$ and $\varvec{Y}$ are given, respectively, by,

• (i) $H_N(\varvec{X}) = \frac{1}{2}\log \vert \varvec{\Sigma }\vert -\frac{1}{2}\log \vert \varvec{\Omega }\vert +{{\,\mathrm{E}\,}}[\log \{2\Phi (\tau W)\}]$, and

• (ii) $H_N(\varvec{Y}) = \frac{1}{2}\log \vert \varvec{\Sigma }\vert +\frac{p}{2}(1+\log 2\pi ) - \frac{1}{2}\log \vert \varvec{\Omega }\vert +\log \Gamma \left( \frac{\nu + p}{2}\right) -\log \Gamma \left( \frac{\nu }{2}\right) - \frac{p}{2}\log (\nu \pi )- \frac{\nu + p}{2} \left\{ \psi \left( \frac{\nu + p}{2}\right) - \psi \left( \frac{\nu }{2}\right) \right\} + {{\,\mathrm{E}\,}}[\log \{2T(\tau W^*; \nu +p)\}]$.

Mardia (1970) introduced one of the popular and commonly used measures of multivariate skewness of an arbitrary p-dimensional random vector $\varvec{Z}$ with mean vector $\varvec{\mu }$ and covariance matrix $\varvec{\Sigma }$. Mardia’s skewness coefficient is defined as,

$$\begin{aligned} \beta _{1,p} = {{\,\mathrm{E}\,}}[\{(\varvec{Z} - \varvec{\mu })^\top \varvec{\Sigma }^{-1}(\varvec{Z} - \varvec{\mu })\}^3], \end{aligned}$$

which can be expressed as $\beta _{1,p} ={{\,\mathrm{tr}\,}}\{\varvec{S}^\top (\varvec{Y})\varvec{S}(\varvec{Y})\}$, where $\varvec{S}(\varvec{Y}) ={{\,\mathrm{E}\,}}(\varvec{Y}\otimes \varvec{Y}^\top \otimes \varvec{Y})$, with $\varvec{Y} =\varvec{\Sigma }^{-1/2} (\varvec{Z} - \varvec{\mu })$ and $\otimes$ denotes the Kronecker product. The following lemmas, extracted from Kim and Mallick (2003), allow us to obtain explicit formulas for $\varvec{S}(\varvec{Y})$. In particular, Fig. 11 leads us to note the interaction between the degrees of freedom and the skewness parameter on the coefficient $\beta _{1,p}$ proposed by Mardia (1970). In fact, as $\nu$ grows, it has less impact on $\beta _{1,p}$.

Lemma 6

If $\varvec{Y} \sim \mathsf {SN}_p(\varvec{0},\varvec{\Omega },\varvec{\gamma })$, then

$$\begin{aligned} \varvec{S}(\varvec{Y}) = \sqrt{2/\pi }[\varvec{\delta }\otimes \varvec{\Omega } +{{\,\mathrm{vec}\,}}(\varvec{\Omega })\varvec{\delta }^\top + (\varvec{I}_p\otimes \varvec{\delta })\varvec{\Omega } - \varvec{\delta }\otimes \varvec{\delta \delta }^\top ], \end{aligned}$$

where $\varvec{\delta } = \varvec{\Omega \gamma }/\sqrt{1 + \tau ^2}$. In addition, if $\varvec{\gamma } = \varvec{0}$, that is $\varvec{Y} \sim \mathsf {N}_p(\varvec{0},\varvec{\Omega })$, then $\beta _{1,p} = 0$.

Lemma 7

If $\varvec{Y} \sim \mathsf {St}_p(\varvec{0},\varvec{\Omega },\varvec{\gamma },\nu )$, then

$$\begin{aligned} \varvec{S}(\varvec{Y}) = \frac{\alpha (\nu )\nu }{\nu - 3}[\varvec{\delta }\otimes \varvec{\Omega } + {{\,\mathrm{vec}\,}}(\varvec{\Omega })\varvec{\delta }^\top + (\varvec{I}_p\otimes \varvec{\delta })\varvec{\Omega } - \varvec{\delta }\otimes \varvec{\delta \delta }^\top ], \end{aligned}$$

where $\alpha (\nu ) = \sqrt{\nu /\pi }\Gamma ((\nu -1)/2)/\Gamma (\nu /2)$ and $\varvec{\delta } = \varvec{\Omega \gamma }/\sqrt{1 + \tau ^2}$. In addition, if $\varvec{\gamma } = \varvec{0}$, that is $\varvec{Y} \sim \mathsf {St}_p(\varvec{0},\varvec{\Omega },\nu )$, then $\beta _{1,p} = 0$.

Fig. 11

Plot of Mardia’s skewness coefficient $\beta _{1,p}$ for the univariate skew t-distribution with $\xi = 0$ and $\Omega = 1$