Coskewness Measure

A Coskewness Measure is a measure of variability between two random variables.

• AKA: Coskewness.
  • …
• Counter-Example(s)
  • Skewness Measure.
  • Kurtosis Measure.
• See: Central Moment, Probability Distribution, Statistical Deviation, Statistical Dispersion.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Coskewness Retrieved 2016-08-21
  • In probability theory and statistics, coskewness is a measure of how much two random variables change together. Coskewness is the third standardized cross central moment, related to skewness as covariance is related to variance. In 1976, Krauss and Litzenberger used it to examine risk in stock market investments. The application to risk was extended by Harvey and Siddique in 2000.

    If two random variables exhibit positive coskewness they will tend to undergo extreme positive deviations at the same time. Similarly, if two random variables exhibit negative coskewness they will tend to undergo extreme negative deviations at the same time.

    Definition: For two random variables X and Y there are two non-trivial coskewness statistics:

[math]\displaystyle{ S(X,X,Y) = \frac{\operatorname{E} \left[(X - \operatorname{E}[X])^2(Y - \operatorname{E}[Y])\right]}{\sigma_X^2 \sigma_Y} }[/math]

and

[math]\displaystyle{ S(X,Y,Y) = \frac{\operatorname{E} \left[(X - \operatorname{E}[X])(Y - \operatorname{E}[Y])^2\right]} {\sigma_X \sigma_Y^2} }[/math]

where E[X] is the expected value of X, also known as the mean of X, and [math]\displaystyle{ \sigma_X }[/math] is the standard deviation of X.

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