Cokurtosis

In probability theory and statistics, cokurtosis is a measure of how much two random variables change together. Cokurtosis is the fourth standardized cross central moment. If two random variables exhibit a high level of cokurtosis they will tend to undergo extreme positive and negative deviations at the same time.

Definition

For two random variables X and Y there are three non-trivial cokurtosis statistics

$K(X,X,X,Y)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])^{3}(Y-\operatorname {E} [Y]){\big ]}} \over \sigma _{X}^{3}\sigma _{Y}},$

$K(X,X,Y,Y)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])^{2}(Y-\operatorname {E} [Y])^{2}{\big ]}} \over \sigma _{X}^{2}\sigma _{Y}^{2}},$

and

$K(X,Y,Y,Y)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])^{3}{\big ]}} \over \sigma _{X}\sigma _{Y}^{3}},$

where E[X] is the expected value of X, also known as the mean of X, and $\sigma _{X}$ is the standard deviation of X.

Properties

• Kurtosis is a special case of the cokurtosis when the two random variables are identical:

$K(X,X,X,X)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])^{4}{\big ]}} \over \sigma _{X}^{4}}={\operatorname {kurtosis} {\big [}X{\big ]}},$

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

$K_{X+Y}={1 \over \sigma _{X+Y}^{4}}{{\big [}\sigma _{X}^{4}K_{X}+4\sigma _{X}^{3}\sigma _{Y}K(X,X,X,Y)+6\sigma _{X}^{2}\sigma _{Y}^{2}K(X,X,Y,Y)+4\sigma _{X}\sigma _{Y}^{3}K(X,Y,Y,Y)+\sigma _{Y}^{4}K_{Y}{\big ]}},$
where $K_{X}$ is the kurtosis of X and $\sigma _{X}$ is the standard deviation of X.

• It follows that the sum of two random variables can have non-zero kurtosis ($K_{X+Y}\neq 0$) even if both random variables are have zero kurtosis in isolation ($K_{X}=0$ and $K_{Y}=0$).
• The cokurtosis between variables X and Y does not depend on the scale on which the variables are expressed. If we are analyzing the relationship between X and Y, the cokurtosis between X and Y will be the same as the cokurtosis between a + bX and c + dY, where a, b, c and d are constants.