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May 31, 2012

Just discovered that the three-way ‘covariance’ I talked about here actually has a name: coskewness. It was hard to find a clear definition so I’m going to use,

coskew(x, y, z) = E((x - E(x))*(y - E(y))*(z - E(z)))

Also found this identity:

coskew(x, y, z) = E(xyz) - E(x)E(y)E(z) - E(x)cov(y,z) - E(y)cov(x,z) - E(z)cov(x,y)

Still not sure about Cauchy-Schwarz for this thing, but here’s a start. coskew and cov are related by,

coskew(x, y, z) = cov((x - E(x))*(y - E(y)), z)
coskew(x, y, z) = cov((y - E(y))*(z - E(z)), x)
coskew(x, y, z) = cov((z - E(z))*(x - E(x)), y)

Therefore, we can Cauchy-Schwarz all of these to get bounds on coskew. However, each of the three versions is not guaranteed to give the same bounds! So far I’ve taken the minimum of the three but I’ve got no proof that there aren’t lesser bounds to find.


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