Selection Constructions for Skew-Elliptical Models

Why Selection Creates Skewness

A symmetric multivariate distribution can produce an asymmetric observed distribution if inclusion depends on a latent variable. For example, imagine that a vector of measurements $X$ is recorded only when an associated latent score $U$ is positive. Even if the joint law of $(X,U)$ is symmetric about its center, the observed $X$-values come from only one side of the selection rule. The resulting distribution can therefore be skewed.

The key calculation is general: the density after selection is the original marginal density multiplied by the probability of satisfying the selection rule, conditional on the observed vector. When the joint law is Gaussian, this probability is a normal tail probability and gives a skew-normal law. When the joint law is Student $t$, it is a conditional Student $t$ tail probability and gives a skew-$t$ law. The two constructions have related structure, but their conditional degrees of freedom and scale behavior differ.

Definition: Let $X$ have density $f_X$, let $U$ be a latent variable jointly distributed with $X$, and let $A$ be a selection event with $P(A)>0$. The selection distribution of $X$ is the conditional distribution of $X$ given $A$. If $P(A\mid X=x)$ is defined through a conditional density, its density is $$ f_{X\mid A}(x)=f_X(x)\frac{P(A\mid X=x)}{P(A)}. $$

This identity follows from the joint probability of $X$ and selection: the density at $x$ together with selection is $f_X(x)P(A\mid X=x)$. Dividing by $P(A)$ normalizes that joint contribution to the conditional density. In the constructions below, $A=\{U>0\}$. If $U$ has a continuous distribution symmetric about zero, then $P(U>0)=1/2$, so the selection multiplier is $2P(U>0\mid X=x)$.

Gaussian Selection Gives a Skew-Normal Density

Let $X$ be $d$-dimensional, let $\xi\in\mathbb{R}^d$, and suppose the joint vector $(X,U)$ is Gaussian with mean $(\xi,0)$ and covariance matrix $$ \Omega= \begin{pmatrix} \Sigma & \delta\\ \delta^T & 1 \end{pmatrix}, $$ where $\Sigma$ is positive definite and $\delta\in\mathbb{R}^d$. This matrix is positive definite precisely when $s=1-\delta^T\Sigma^{-1}\delta>0$. Define $y=x-\xi$ and $a(x)=\delta^T\Sigma^{-1}y$. The Gaussian conditional law is $U\mid X=x\sim N(a(x),s)$, where the second argument is the variance. Consequently, $P(U>0\mid X=x)=\Phi(a(x)/\sqrt{s})$.

Result: Under the Gaussian joint model above, the density of $X\mid U>0$ is $$ f_{X\mid U>0}(x) = 2\phi_d(x;\xi,\Sigma) \Phi\!\left( \frac{\delta^T\Sigma^{-1}(x-\xi)} {\sqrt{1-\delta^T\Sigma^{-1}\delta}} \right), $$ where $\phi_d(\,\cdot\,;\xi,\Sigma)$ is the $d$-variate Gaussian density with mean $\xi$ and covariance $\Sigma$, and $\Phi$ is the standard normal CDF.

To verify the conditional Gaussian law, the Schur complement of $\Sigma$ in $\Omega$ is $s$. The conditional mean of $U$ given $X=x$ is $\delta^T\Sigma^{-1}(x-\xi)$, and its conditional variance is $s$. These are the standard block-Gaussian conditional parameters. Since $U$ is centered and continuous, $P(U>0)=1/2$; substituting its conditional tail probability into the selection identity proves the displayed density.

Worked Example: A Univariate Gaussian Selection Density

Suppose $X$ and $U$ are jointly Gaussian with mean zero and covariance matrix $$ \begin{pmatrix}1&0.6\\0.6&1\end{pmatrix}. $$ Find the density of $X\mid U>0$ at $x=1$.

Here $\Sigma=1$, $\delta=0.6$, and $s=1-0.6^2=0.64$, so $\sqrt{s}=0.8$. The selection formula gives $$ f_{X\mid U>0}(1) = 2\phi(1)\Phi\!\left(\frac{0.6(1)}{0.8}\right) = 2\phi(1)\Phi(0.75). $$

Using $\phi(1)\approx0.2419707$ and $\Phi(0.75)\approx0.7733726$, $$ f_{X\mid U>0}(1) \approx 2(0.2419707)(0.7733726) \approx 0.3743. $$ The value exceeds the unselected density $\phi(1)\approx0.2420$: positive values of $X$ make $U>0$ more likely because their correlation is positive.

Student t Selection Gives a Skew-t Density

For the Student $t$ construction, use the same block form for a scale matrix: $$ \Omega= \begin{pmatrix} \Sigma & \delta\\ \delta^T & 1 \end{pmatrix}, \qquad s=1-\delta^T\Sigma^{-1}\delta>0. $$ Assume $(X,U)$ has a $(d+1)$-variate Student $t$ distribution with degrees of freedom $\nu>0$, location $(\xi,0)$, and scale matrix $\Omega$. Thus $X$ marginally has a $d$-variate Student $t$ distribution with degrees of freedom $\nu$, location $\xi$, and scale matrix $\Sigma$. For Student $t$ distributions, a scale matrix is not generally a covariance matrix; the covariance exists only when the degrees of freedom exceed $2$, and then equals the scale matrix multiplied by $\nu/(\nu-2)$.

Result: Write $y=x-\xi$, $Q(x)=y^T\Sigma^{-1}y$, and $a(x)=\delta^T\Sigma^{-1}y$. Under the joint Student $t$ model, $$ U\mid X=x \sim t_{\nu+d}\!\left( a(x),\, \frac{s\{\nu+Q(x)\}}{\nu+d} \right), $$ where the second argument is the scalar scale squared (the univariate scale-matrix parameter). Therefore, $$ f_{X\mid U>0}(x) = 2 f_{t_{\nu,d}}(x;\xi,\Sigma) T_{\nu+d}\!\left( \frac{a(x)} {\sqrt{s\{\nu+Q(x)\}/(\nu+d)}} \right). $$ Here $T_m$ is the CDF of a standard univariate Student $t$ distribution with $m$ degrees of freedom, and $f_{t_{\nu,d}}$ is the $d$-variate Student $t$ density.

A derivation makes the scale parameter unambiguous. The joint $t$ density is proportional to $|\Omega|^{-1/2}[1+z^T\Omega^{-1}z/\nu]^{-(\nu+d+1)/2}$, where $z=(y^T,u)^T$. Block-matrix inversion, or completing the square in the quadratic form, gives $$ z^T\Omega^{-1}z = Q(x)+\frac{\{u-a(x)\}^2}{s}. $$ It follows that the part of the joint density depending on $u$, with $x$ fixed, is proportional to $$ \left[ 1+\frac{\{u-a(x)\}^2}{s\{\nu+Q(x)\}} \right]^{-(\nu+d+1)/2}. $$ A univariate $t_m$ density with location $a$ and scale squared $b^2$ has its variable-dependent factor $[1+(u-a)^2/(mb^2)]^{-(m+1)/2}$. Matching exponents gives $m=\nu+d$; matching the bracket gives $(\nu+d)b^2=s\{\nu+Q(x)\}$. Thus $b^2=s\{\nu+Q(x)\}/(\nu+d)$, proving the conditional law. Its conditional selection probability is $T_{\nu+d}(a(x)/b)$. Finally, $U$ is symmetric about zero, so $P(U>0)=1/2$; the selection identity supplies the factor $2$ in the density.

Worked Example: A Univariate Skew-t Selection Probability

Let $(X,U)$ have a bivariate Student $t$ distribution with $\nu=5$, location zero, and scale matrix $$ \begin{pmatrix}1&0.5\\0.5&1\end{pmatrix}. $$ Find $P(U>0\mid X=1)$, then express the selected density of $X$ at $1$.

Here $d=1$, $s=1-0.5^2=0.75$, $Q(1)=1$, and $a(1)=0.5$. The conditional degrees of freedom are $\nu+d=6$, and the conditional scale squared is $$ \frac{0.75(5+1)}{6}=0.75. $$ The scalar scale is therefore $\sqrt{0.75}$, not $0.75$. Hence $$ P(U>0\mid X=1) = T_6\!\left(\frac{0.5}{\sqrt{0.75}}\right) = T_6(1/\sqrt{3}) \approx 0.7077. $$

The marginal density of $X$ at $1$ is $$ f_{t_{5,1}}(1) = \frac{\Gamma(3)}{\Gamma(5/2)\sqrt{5\pi}} \left(1+\frac{1}{5}\right)^{-3} \approx 0.2197. $$ Therefore the density after selection is $$ f_{X\mid U>0}(1) = 2(0.2197)(0.7077) \approx 0.3110. $$ The selection multiplier uses the conditional probability $0.7077$; it is not the conditional $t$ density at zero.

Worked Example: Checking a Two-Dimensional t Construction

Take $\nu=7$, $\xi=(0,0)^T$, $\Sigma=\operatorname{diag}(4,1)$, and $\delta=(1,0.3)^T$. Check that the construction is valid and derive the selected density at $x=(2,-1)^T$.

The matrix $\Sigma$ is positive definite, and $$ \delta^T\Sigma^{-1}\delta = (1,0.3) \begin{pmatrix}1/4&0\\0&1\end{pmatrix} \begin{pmatrix}1\\0.3\end{pmatrix} = 0.25+0.09=0.34. $$ Thus $s=1-0.34=0.66>0$, so the block scale matrix is positive definite. At $x=(2,-1)^T$, $$ Q(x)=\frac{2^2}{4}+(-1)^2=2, \qquad a(x)=\frac{1(2)}{4}+0.3(-1)=0.2. $$ The conditional degrees of freedom are $7+2=9$, and the conditional scale squared is $$ \frac{0.66(7+2)}{9}=0.66. $$ Accordingly, the standardized selection threshold is $0.2/\sqrt{0.66}\approx0.2462$, and $$ f_{X\mid U>0}(x) = 2 f_{t_{7,2}}(x;0,\Sigma) T_9(0.2462). $$ For completeness, $|\Sigma|=4$, and the marginal density at this point is $$ f_{t_{7,2}}(x;0,\Sigma) = \frac{\Gamma(9/2)}{\Gamma(7/2)(7\pi)\sqrt{4}} \left(1+\frac{2}{7}\right)^{-9/2} = \frac{1}{4\pi}\left(\frac{9}{7}\right)^{-9/2}. $$ These quantities specify the selected density without confusing the conditional scale squared $0.66$ with the scalar scale $\sqrt{0.66}$.

What the Two Constructions Share—and What They Do Not

Both models use the same selection identity and the same block-matrix validity condition $s>0$. In each, $\delta$ controls how the observed vector predicts the latent selection variable: $a(x)=\delta^T\Sigma^{-1}(x-\xi)$ is the conditional location of $U$. When $a(x)$ is large and positive, selection is more likely; when it is negative, selection is less likely. If $\delta=0$, then $a(x)=0$ for every $x$, the conditional selection probability is $1/2$, and conditioning on $U>0$ leaves the marginal law of $X$ unchanged.

The crucial difference is in the conditional selection probability. In the Gaussian model, the conditional variance $s$ is constant, so the argument of $\Phi$ is a fixed linear function of $x-\xi$. In the Student $t$ model, the conditional degrees of freedom increase from $\nu$ to $\nu+d$, and the conditional scale squared depends on the Mahalanobis distance $Q(x)$. The tail probability is therefore not obtained by merely replacing $\Phi$ with a $t$ CDF in the Gaussian formula: both the CDF degrees of freedom and its scale argument change.

Common Pitfall: Scale Versus Scale Squared

A frequent error is to state that $U\mid X=x$ has scalar $t$ scale $s\{\nu+Q(x)\}/(\nu+d)$ and then divide its location by the square root of that quantity. Those two statements contradict each other. The displayed expression is the scale squared; the scalar scale used to standardize the threshold is its square root. In the univariate example, the conditional scale squared is $0.75$, while the scalar scale is $\sqrt{0.75}$. Thus the CDF argument is $0.5/\sqrt{0.75}$, not $0.5/0.75$.

A second check is normalization: the selection density must integrate to one. Indeed, integrating $f_X(x)P(U>0\mid X=x)$ over $x$ gives $P(U>0)$ by the law of total probability; dividing by $P(U>0)$ therefore normalizes the result. This provides a useful safeguard against omitting the factor $2$ in the centered symmetric constructions.

Key takeaway: Selection skewness is encoded by a conditional event probability multiplying the original marginal density. Gaussian latent laws yield a normal-CDF factor; Student $t$ latent laws yield a $t$-CDF factor with updated degrees of freedom and a distance-dependent scale.

Check Your Understanding

Use the selection identity and the conditional laws above to answer these questions.

Why does selection on $U>0$ contribute a factor of $2$ when $U$ is centered, continuous, and symmetric?
For a Gaussian joint model, what condition on $\delta$ and $\Sigma$ ensures that the block covariance matrix is positive definite?
In the Student $t$ construction, what are the conditional degrees of freedom of $U\mid X=x$ when $X$ has dimension $d$ and the joint degrees of freedom are $\nu$?
In the univariate Student $t$ example, distinguish the conditional scale squared from the scalar scale.
What happens to the selected marginal distribution of $X$ if $\delta=0$?