Cokurtosis
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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.[1] 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 [1][2]
![{\displaystyle K(X,X,X,Y)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])^{3}(Y-\operatorname {E} [Y]){\big ]}} \over \sigma _{X}^{3}\sigma _{Y}},}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2f10a5a4c95549714ee5e66ce3c869709e8e2722)
![{\displaystyle 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}},}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/57a1a9a04b08c6239888069e405c65ac26842cd2)
and
![{\displaystyle K(X,Y,Y,Y)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])^{3}{\big ]}} \over \sigma _{X}\sigma _{Y}^{3}},}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bd1085c4784002797298f5e9d7f1705457205ed4)
where E[X] is the expected value of X, also known as the mean of X, and is the standard deviation of X.
Properties
- Kurtosis is a special case of the cokurtosis when the two random variables are identical:
![{\displaystyle K(X,X,X,X)={\operatorname {E} {{\big [}(X-\operatorname {E} [X])^{4}{\big ]}} \over \sigma _{X}^{4}}={\operatorname {kurtosis} {\big [}X{\big ]}},}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/cf5ceee06a0e7843e26dd089cfe9141493bf708e)
- For two random variables, X and Y, the kurtosis of the sum, X + Y, is
- where is the kurtosis of X and is the standard deviation of X.
![{\displaystyle {\begin{aligned}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 ]},\end{aligned}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/0021482871a17a742370f6e0f113f0af00875ddc)
- It follows that the sum of two random variables can have non-zero kurtosis () even if both random variables are have zero kurtosis in isolation ( and ).
- 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.
See also
References
- ^ a b Miller, Michael B. (2014). Mathematics and Statistics for Financial Risk Management (2nd ed.). Hoboken, New Jersey: John Wiley & Sons, Inc. pp. 53–56. ISBN 978-1-118-75029-2.
- ^ Meucci, Attilio (2005). Risk and Asset Allocation. Berlin: Springer-Verlag. pp. 58–59. ISBN 978-3642009648.
Further reading
- Ranaldo, Angelo; Laurent Favre (2005). "How to Price Hedge Funds: From Two- to Four-Moment CAPM". UBS Research Paper.
- Christie-David, R.; M. Chaudry (2001). "Coskewness and Cokurtosis in Futures Markets". Journal of Empirical Finance. 8 (1): 55–81.
Volume 8, Issue 1, March 2001, Pages 55–81
Category:Algebra of random variables Category:Theory of probability distributions