Moments of the Multivariate Skew-Normal

Why the Latent Representation Is Useful

The skew-normal distribution introduces asymmetry through a selection step: a component of a jointly normal vector is required to be positive. That construction is more than a way to define the model. It makes its first two moments accessible by separating the selected latent variable from a normal residual. The resulting formulas also show why a parameter called “shape” affects both the mean and covariance, as well as the marginal skewness.

We derive these quantities directly, starting from the latent-normal representation used for Azzalini’s multivariate skew-normal distribution. Throughout, the shape vector describes association with the selected latent variable; it is not itself the vector of marginal skewnesses.

The conditioning notation means that we consider the conditional distribution of \(\xi+\Delta Z\) given the event \(T>0\). The block covariance condition ensures a nondegenerate joint normal distribution. In particular, it permits the decomposition \(Z=\delta T+\varepsilon\), where \(\varepsilon\) is normal, independent of \(T\), with covariance \(\Omega-\delta\delta^\top\).

Moments from the Selected Normal Variable

Conditioning a standard normal variable to be positive gives a half-normal variable. Write \(c=\sqrt{2/\pi}\). The moments needed below follow directly by symmetry and integration:

For example, \(E[T\mid T>0]=E[|T|]=\sqrt{2/\pi}\). The second moment is unchanged because \(T^2\) is even and the positive and negative half-lines contribute equally. For the third moment, \(E[T^3\mid T>0]=2\int_0^\infty t^3\phi(t)\,dt=2c\), where \(\phi\) is the standard normal density.

Proof. Joint normality gives \(Z=\delta T+\varepsilon\), where \(\varepsilon\) is independent of \(T\), has mean zero, and has covariance \(\Omega-\delta\delta^\top\). Independence is preserved when conditioning on \(T>0\): the event depends on \(T\), not on \(\varepsilon\). Thus

For the covariance, the two terms \(\delta T\) and \(\varepsilon\) have zero conditional cross-covariance, by independence and \(E[\varepsilon]=0\). Therefore

The first equality uses \(\operatorname{Var}(T\mid T>0)=1-c^2\), from the first two half-normal moments. Applying the affine transformation \(X=\xi+\Delta Z\) gives the stated formulas. \(\square\)

Worked Examples: Applying the Mean and Covariance Formulas

Marginal Standardized Skewness

Assume first that \(\Omega_{ii}=1\), as in the common correlation-matrix parameterization. The \(i\)th component has variance \(\Delta_{ii}^2(1-c^2\delta_i^2)\). Its standardized skewness is its third central moment divided by the variance to the power \(3/2\). To derive that third moment, the half-normal variable’s third central moment is

Write \(Z_i=\delta_iT+\varepsilon_i\), with \(\varepsilon_i\) independent of \(T\), centered normal, and with variance \(1-\delta_i^2\). After centering \(Z_i\), expand the cube of \(\delta_i(T-c)+\varepsilon_i\). The term involving \((T-c)^3\) has expectation \(\delta_i^3 E[(T-c)^3\mid T>0]\). The mixed term with \((T-c)\varepsilon_i^2\) has expectation zero because \(E[T-c\mid T>0]=0\); the other mixed term and \(E[\varepsilon_i^3]\) are also zero. Therefore the third central moment of \(Z_i\) is \(\delta_i^3 c(4-\pi)/\pi\).

To see the formula, divide the third central moment of \(Z_i\) by \((1-c^2\delta_i^2)^{3/2}\). The identity \(c(4-\pi)/\pi=c^3(4-\pi)/2\) then gives the displayed expression. The scale \(\Delta_{ii}\) multiplies the third central moment by \(\Delta_{ii}^3\) and the variance to the \(3/2\) power by the same factor, so it cancels.

For a general positive \(\Omega_{ii}\), replace \(\delta_i\) in the formula by \(\delta_i/\sqrt{\Omega_{ii}}\). This follows by applying the unit-variance result to \(Z_i/\sqrt{\Omega_{ii}}\). The block covariance condition ensures \(c^2\delta_i^2<\Omega_{ii}\), so the marginal variance is positive and the denominator is defined.

How Shape Changes Marginal Asymmetry

For unit marginal variance in \(\Omega\), the formula depends on \(\delta_i\) through the signed quantity \(c\delta_i/\sqrt{1-c^2\delta_i^2}\). Its denominator is positive, and the outer factor \((4-\pi)/2\) is positive. Thus the sign of marginal skewness is the sign of \(\delta_i\); when \(\delta_i=0\), the marginal is normal and its skewness is zero. As \(|\delta_i|\) approaches its admissible limit of \(1\), the variance \(1-c^2\delta_i^2\) remains positive, but the magnitude of the skewness increases.

This relationship is useful when interpreting a fitted shape vector: its component signs indicate the directions of marginal asymmetry, while its magnitudes affect both the conditional mean and variance. The covariance also contains off-diagonal terms \(-c^2\delta_i\delta_j\) before scaling. Consequently, selection can alter dependence as well as marginal locations and spreads.

Common Mistakes

• Using the unconditioned covariance as the model covariance. The unconditioned \(Z\) has covariance \(\Omega\), but conditioning on \(T>0\) changes it to \(\Omega-c^2\delta\delta^\top\). The correction follows from subtracting the square of the shifted conditional mean.
• Forgetting both scale factors on an off-diagonal covariance. Under \(X=\xi+\Delta Z\), entry \(i,j\) is multiplied by \(\Delta_{ii}\Delta_{jj}\), not just one of the two factors. In the bivariate example, the off-diagonal \(-2/(3\pi)\) becomes \(-4/(3\pi)\).
• Calling \(\delta_i\) the marginal skewness. It is a latent association parameter. Standardized skewness is a nonlinear function of it, as shown above.
• Dropping a square root or a factor when evaluating the skewness formula. For example, in the last worked example the ratio simplifies to \(1/\sqrt{4\pi-1}\); cubing it gives the power \(3/2\) in the denominator. The resulting value is approximately \(0.01091\), not \(0.01118\).