Moments and Tail Behavior of Skew-T Models

Why Moments and Tails Need Separate Attention

A skew-\(t\) model combines asymmetry with a random scale. These features affect different aspects of its distribution: the skewness parameter can shift the mean and make the two marginal tails unequal, while the degrees of freedom determine whether moments exist and how slowly the tails decay. A finite location parameter alone guarantees neither a finite mean nor a finite covariance.

We use the scale-mixture construction to make these distinctions precise. In particular, we will keep track of the scale matrix in the construction separately from the covariance matrix of the resulting vector.

Scale-Mixture Definition and Moment Formulas

Let \(\nu>0\), let \(\boldsymbol{\mu}\in\mathbb{R}^p\), let \(\boldsymbol{\delta}\in\mathbb{R}^p\), and let \(\boldsymbol{\Sigma}\) be a positive-definite matrix such that \(\boldsymbol{\Sigma}-\boldsymbol{\delta}\boldsymbol{\delta}'\) is positive definite. Choose a matrix \(A\) satisfying \(AA'=\boldsymbol{\Sigma}-\boldsymbol{\delta}\boldsymbol{\delta}'\). Let \(U_0\sim N(0,1)\), \(\mathbf{U}\sim N_p(\mathbf{0},I)\), and \(W\sim\chi^2_\nu\) be mutually independent. Define

This tutorial uses this construction as the definition of the skew-\(t\) model. The vector \(\mathbf{Z}\) is skew-normal, and the independent positive scale \(T\) supplies the heavy tails. Write \(q=\sqrt{2/\pi}\). Since \(E[|U_0|]=q\) and \(\operatorname{Var}(|U_0|)=1-q^2\),

For the last identity, add the covariance matrix to the outer product of the mean: \(\boldsymbol{\Sigma}-q^2\boldsymbol{\delta}\boldsymbol{\delta}'+q^2\boldsymbol{\delta}\boldsymbol{\delta}'=\boldsymbol{\Sigma}\). Thus \(\boldsymbol{\Sigma}\) is the raw second-moment matrix of the skew-normal component, not its covariance.

Proof. The chi-square density is \(f_W(w)=w^{\nu/2-1}e^{-w/2}/(2^{\nu/2}\Gamma(\nu/2))\) for \(w>0\). For \(r<\nu\), direct integration gives

The integral is finite exactly when \(\nu>r\). Independence of \(T\) and \(\mathbf{Z}\) therefore gives \(E[\mathbf{Y}]=\boldsymbol{\mu}+E[T]E[\mathbf{Z}]\) when \(\nu>1\), yielding the stated mean. For \(\nu>2\), \(E[T^2]=\nu/(\nu-2)\), so

Substitution gives the covariance formula. For necessity of the thresholds, every nondegenerate marginal \(Z_j\) has a finite, strictly positive first absolute moment and a finite, strictly positive second moment. Independence and Tonelli's theorem give \(E[|TZ_j|]=E[T]E[|Z_j|]\) and \(E[T^2Z_j^2]=E[T^2]E[Z_j^2]\), including when the right-hand sides are infinite. The scale moments diverge when \(\nu\leq1\) and \(\nu\leq2\), respectively. Hence the stated thresholds are necessary as well as sufficient. \(\square\)

Worked Bivariate Moment Calculation

The formula for the covariance combines a scale inflation term with a skewness correction. The correction is an outer product, so it affects diagonal and off-diagonal entries together.

Marginal Tail Rates and Skewness

The random scale also gives a direct way to describe marginal tail behavior. For a fixed coordinate \(j\), write \(Z_j\) for the corresponding component of \(\mathbf{Z}\), and let \(x_+=\max(x,0)\). The scale \(T=\sqrt{\nu/W}\) has a power tail. Indeed, as \(t\to\infty\),

To see the constant, integrate the chi-square density near zero: \(P(W\leq y)\sim y^{\nu/2}/(2^{\nu/2}\Gamma(\nu/2+1))\) as \(y\downarrow0\), and set \(y=\nu/t^2\). The resulting marginal tail constants depend on the positive and negative moments of the skew-normal component.

Proof. The variable \(Z_j\) is independent of \(T\), and it has finite absolute moments of every positive order because it is a linear combination of a half-normal and a normal variable. Choose \(C<\infty\) such that \(P(T>t)\leq Ct^{-\nu}\) for every \(t>0\). Such a constant exists: the tail asymptotic supplies the bound for sufficiently large \(t\), and for \(t\) in a bounded interval the probability is at most \(1\), while \(t^{-\nu}\) is bounded below by a positive number. Conditioning on \(Z_j\) gives

For each fixed \(z>0\), the tail asymptotic implies \(x^\nu P(T>x/z)\to k_\nu z^\nu\). Also, for every \(x>0\) and \(z>0\), \(x^\nu P(T>x/z)\leq C z^\nu\). The bound is integrable when evaluated at \(Z_j\), since \(E[(Z_j)_+^\nu]<\infty\). Dominated convergence now yields

Applying the same argument to \(-Z_j\) proves the lower-tail formula. The skew-normal marginal has positive probability on both sides of zero in a nondegenerate model, so both constants are positive. \(\square\)

Interpretation and a Common Pitfall

The marginal tail result separates two effects that are easy to conflate. Lower degrees of freedom mean a smaller power exponent and hence slower polynomial tail decay. Skewness changes the allocation of tail mass between the positive and negative sides through the two positive-part moments. It does not change the exponent in this scale-mixture model.

Common mistake: It is tempting to call \(\boldsymbol{\Sigma}\) the covariance of \(\mathbf{Y}\), or to infer that the covariance is just \(\nu/(\nu-2)\) times \(\boldsymbol{\Sigma}\). The first claim ignores the random scale and skewness; the second ignores the centering term. The actual covariance, when it exists, is \(\nu\boldsymbol{\Sigma}/(\nu-2)-c_\nu^2q^2\boldsymbol{\delta}\boldsymbol{\delta}'\). Another error is to claim a finite covariance whenever the location and scale parameters are finite: the necessary condition is \(\nu>2\).