Scale-Mixture Constructions of Skew-T Distributions

Why Mix a Skew-Normal Vector Over a Scale?

A skew-normal model can represent asymmetry, while a Student \(t\) model represents heavier tails than a Gaussian model. A positive scale mixture combines these features: conditional on the scale, the vector has a skew-normal distribution; variation in the scale allows occasional observations to be much more dispersed. The resulting model is a multivariate skew-\(t\) distribution.

The construction is useful because it separates two roles. The skew-normal shape parameter controls the skewing mechanism, while the random scale controls tail thickness. The derivation also shows why the skew-\(t\) density contains a Student \(t\) distribution function rather than a Gaussian distribution function.

Definition of the Scale Mixture

Let \(p\) be the dimension, let \(\xi\in\mathbb{R}^p\), and let \(\Omega\) be a positive definite \(p\times p\) matrix. Write \(\omega=\operatorname{diag}(\sqrt{\Omega_{11}},\ldots,\sqrt{\Omega_{pp}})\), and let \(\alpha\in\mathbb{R}^p\) be a shape vector. The multivariate skew-normal density in the parameterization used here is

where \(\phi_p(\,\cdot\,;\xi,\Omega)\) is the \(N_p(\xi,\Omega)\) density and \(\Phi\) is the standard normal distribution function. In particular, \(\Omega\) is the scale matrix of the underlying Gaussian kernel; when \(\alpha\ne 0\), it need not equal the covariance matrix of the skew-normal vector.

For the conditional density, the diagonal scale matrix associated with \(\Omega/w\) is \(\omega/\sqrt w\). Thus the conditional skewing argument is \(\sqrt w\,\alpha^\top\omega^{-1}(x-\xi)\). Define

Deriving the Marginal Density

The key calculation is an expectation of a normal distribution function over a Gamma variable. It converts the conditional skewing factor into a Student \(t\) distribution function.

Proof. Let \(U=\lambda W\), so \(U\sim\chi^2_m\), and let \(N\sim N(0,1)\) be independent of \(U\). By conditioning on \(W\),

The ratio \(N/\sqrt{U/m}\) has a standard \(t_m\) distribution by the defining normal-over-independent-chi-squared representation. The displayed probability is therefore the stated \(t_m\) distribution function. The division is valid because \(U>0\) with probability one. This argument applies to negative as well as positive \(c\), since the denominator is positive. \(\square\)

We now integrate the conditional skew-normal density. The density of \(X\) is the integral of that conditional density against the Gamma density of \(W\). In the integrand, separate the Gaussian kernel from the skewing factor. The product of the prior density of \(W\) and the Gaussian kernel is proportional to

Consequently, after normalizing this factor, the conditional distribution of \(W\) given \(x\) under the Gaussian-kernel part of the mixture is \(\operatorname{Gamma}((\nu+p)/2,(\nu+Q)/2)\). This is a calculation within the density integral; it does not assert that the full skew-\(t\) posterior distribution of \(W\) has this form.

Derivation. The Gaussian-kernel part of the mixture integral is the multivariate \(t\) density displayed in the theorem. This follows by integrating the Gaussian density against the Gamma density: the determinant contributes \(w^{p/2}\), the Gaussian exponential contributes \(\exp(-wQ/2)\), and the Gamma density contributes \(w^{\nu/2-1}\exp(-\nu w/2)\). The resulting Gamma integral has shape \((\nu+p)/2\) and rate \((\nu+Q)/2\), giving the stated normalizing constant and power.

The remaining factor is the expectation of \(\Phi(z\sqrt W)\) with respect to that normalized Gamma density. Apply the lemma with \(m=\nu+p\), \(\lambda=\nu+Q\), and \(c=z\). This gives \(T_{\nu+p}(z\sqrt{(\nu+p)/(\nu+Q)})\). Multiplying by the factor \(2\) in the conditional skew-normal density proves the theorem. Since the result was obtained by integrating conditional probability densities, it is normalized and nonnegative. \(\square\)

How the Mixing Variable Controls Tail Thickness

The Gamma mixing variable has mean \(E[W]=1\) and variance \(\operatorname{Var}(W)=2/\nu\), as follows from the mean and variance formulas for a Gamma distribution with shape and rate \(\nu/2\). Hence increasing \(\nu\) concentrates the scale more tightly around \(1\), while smaller \(\nu\) permits greater scale variation. In the limiting regime as \(\nu\) grows, the scale becomes increasingly concentrated near one; at finite \(\nu\), small values of \(W\) can produce much broader conditional kernels.

The density derivation makes the tail mechanism precise at the level of the base kernel: mixing a Gaussian kernel over \(W\) gives a multivariate \(t\) kernel with polynomial rather than Gaussian decay. The skewing multiplier is \(2T_{\nu+p}(\cdot)\), which lies between \(0\) and \(2\). Along any fixed direction away from \(\xi\), its argument has a finite limit, so the multiplier approaches a positive finite value; it changes directional density weights but does not remove the polynomial tail generated by the scale mixture. The degrees of freedom \(\nu\) determine the polynomial order of the \(t\) kernel.

A common mistake is to confuse the Gamma variable used in the mixture with a chi-squared variable. For the prior mixing law, \(U=\nu W\sim\chi^2_\nu\), not \(2\nu W\). In the density calculation, the relevant posterior-kernel law is different: if \(W\sim\operatorname{Gamma}(m/2,\lambda/2)\), then \(\lambda W\sim\chi^2_m\). Keeping the rate parameter explicit prevents incorrect rescaling in the normal-over-chi-squared identity.