ⓘ Coskewness
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.
1. Definition
For three random variables X, Y and Z, the nontrivial coskewness statistic is defined as:
S X, Y, Z = E is the expected value of X, also known as the mean of X, and σ X {\displaystyle \sigma _{X}} is the standard deviation of X.2. Properties
Skewness is a special case of the coskewness when the three random variables are identical:
S X, X, X = E },}where S X is the skewness of X and σ X {\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 coskewness 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 coskewness between X and Y will be the same as the coskewness between a + bX and c + dY, where a, b, c, and d are constants.
3. Example
Let X be standard normally distributed and Y be the distribution obtained by setting X = Y 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
f X, Y x, y = e − x 2 / 2 π H − x δ x − y + 2 H x H y e − y 2 / 2 π) {\displaystyle f_{X,Y}x,y={\frac {e^{x^{2}/2}}{\sqrt {2\pi }}}\leftHx\delta xy+2HxHy{\frac {e^{y^{2}/2}}{\sqrt {2\pi }}}\right)}where H x is the Heaviside step function and δx is the Dirac delta function. The third moments are easily calculated by integrating with respect to this density:
S X, X, Y = S X, Y, Y = − 1 2 π ≈ − 0.399 {\displaystyle SX,X,Y=SX,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
cov X, Y = 1 2 + 1 π {\displaystyle \operatorname {cov} X,Y={\frac {1}{2}}+{\frac {1}{\pi }}}from which it follows that the standard deviation of their sum is
σ X + Y = 3 + 2 π {\displaystyle \sigma _{X+Y}={\sqrt {3+{\frac {2}{\pi }}}}}Using the skewness sum formula above, we have
S X + Y = − 3 2 π 2 + 3 π 3 / 2 ≈ − 0.345 {\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:
f X + Y u = e − u 2 / 8 2 π H − u + e − u 2 / 4 π erf u 2 H u {\displaystyle f_{X+Y}u={\frac {e^{u^{2}/8}}{2{\sqrt {2\pi }}}}Hu+{\frac {e^{u^{2}/4}}{\sqrt {\pi }}}\operatorname {erf} \left{\frac {u}{2}}\rightHu} are moments involving multiple variables. Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple
 all possible quantile skewness measures. Mathematics portal Bragg peak Coskewness Shape parameters Skew normal distribution Skewness risk Susan Dean, Barbara
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