Coskewness

Properties

Skewness is a special case of the coskewness when the three random variables are identical:

${\displaystyle S(X,X,X)=\frac{\mathrm{E}\left[(X-\mathrm{E}[X]{)}^3\right]}{{\sigma }_X^3}=\mathrm{skewness}[X],}$

For two random variables, X and Y, the skewness of the sum, X + Y, is

${\displaystyle S_{X+Y}=\frac{1}{{\sigma }_{X+Y}^3}\left[{\sigma }_X^3{S}_X+3{\sigma }_X^2{\sigma }_{Y}S(X,X,Y)+3{\sigma }_X{\sigma }_Y^{2}S(X,Y,Y)+{\sigma }_Y^3{S}_Y\right],}$

where S_X is the skewness of X and ${\displaystyle {\sigma }_X}$ is the standard deviation of X. It follows that the sum of two random variables can be skewed (S_X+Y ≠ 0) even if both random variables have zero skew in isolation (S_X = 0 and S_Y = 0).

The standardized rank coskewness RS(X, Y, Z) satisfies the following properties:

(1) −1 ≤ RS(X, Y, Z) ≤ 1.

(2) The upper bound of 1 is obtained by the copula given in (3.3) in Bernard, Chen, Rüschendorf and Vanduffel (2023). The lower bound of −1 is obtained by the copula (3.5) in the same paper.

(3) It is invariant under strictly increasing transformations, i.e., when f_i, i = 1, 2, 3, are arbitrary strictly increasing functions, RS(X, Y, Z) = RS(f₁ (X), f₂ (Y), f₃ (Z)).

(4) RS(X, Y, Z) = 0 if X, Y and Z are independent.

Example

Let X be standard normally distributed and Y be the distribution obtained by setting X=Y whenever X<0 and drawing Y independently from a standard half-normal distribution whenever X>0. In other words, X and Y are both standard normally distributed with the property that they are completely correlated for negative values and uncorrelated apart from sign for positive values. The joint probability density function is

${\displaystyle f_{X,Y}(x,y)=\frac{e^{-x^2/2}}{\sqrt{2\pi }}\left(H(-x)\delta (x-y)+2H(x)H(y)\frac{e^{-y^2/2}}{\sqrt{2\pi }}\right)}$

where H(x) is the Heaviside step function and δ(x) is the Dirac delta function. The third moments are easily calculated by integrating with respect to this density:

${\displaystyle S(X,X,Y)=S(X,Y,Y)=-\frac{1}{\sqrt{2\pi }}\approx -0.399}$

Note that although X and Y are individually standard normally distributed, the distribution of the sum X+Y is significantly skewed. From integration with respect to density, we find that the covariance of X and Y is

${\displaystyle \mathrm{cov}(X,Y)=\frac{1}{2}+\frac{1}{\pi }}$

from which it follows that the standard deviation of their sum is

${\displaystyle {\sigma }_{X+Y}=\sqrt{3+\frac{2}{\pi }}}$

Using the skewness sum formula above, we have

${\displaystyle S_{X+Y}=-\frac{3\sqrt{2}\pi }{(2+3\pi {)}^{3/2}}\approx -0.345}$

This can also be computed directly from the probability density function of the sum:

${\displaystyle f_{X+Y}(u)=\frac{e^{-u^2/8}}{2\sqrt{2\pi }}H(-u)+\frac{e^{-u^2/4}}{\sqrt{\pi }}\mathrm{erf}\left(\frac{u}{2}\right)H(u)}$

Bernard, Chen, Rüschendorf and Vanduffel (2023) found risk bounds on coskewness for some popular marginal distributions as shown in the following table.

where ${\displaystyle \mathrm{\Gamma }(x)}$ is the gamma function.