Sunday, August 21, 2016

Covariance formula with CDF (Hoeffding's Covariance Identity)

{\displaystyle \operatorname {cov} (X,Y)=\int _{\mathbb {R} }\int _{\mathbb {R} }F_{XY}(x,y)-F_{X}(x)F_{Y}(y)dxdy}

A complete proof of above lemma can be found on page 241 (Lemma 7.27) of Quantitative Risk Management: Concepts, Techniques and Tools.

Hint: 2\(cov(X_1, X_2) = E[(X_1-\tilde{X_1})(X_2-\tilde{X_2})]\),
where \( (\tilde{X_1}, \tilde{X_2}) \) is an independent copy with the same joint distribution function as  \( (X_1, X_2) \).

Link to MathJax

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