A Surprising Covariance Involving the Minimum Of Multivariate Normal Variables
Aformula is provided for the covariance of X 1 with the minimum of the multivariate normal vector (X1, X2, … Xn ). The resulting expression has an intuitive interpretation as the weighted average of then covariances of X1 with X1, X2, … Xn , where the ith weight equals the probability that X1 is the minimum. The formula is surprising for several reasons. First, results involving extrema of order statistics in the presence of correlation are usually much more complex. Second, anattempt at adirect proof runs into difficulties involving the nontrivial distinction between a conditional expectation and an ordinary covariance. Finally, although these difficulties are genuine on a term-by-term basis, they cancel out when weighted and combined. The geometric interpretation in n-dimensional space is that although the vector of covariances of X1 with (X1, X2, … Xn ) is in general different from the vector of appropriate conditional expectations, these vectors always have the same projection on to the vector of prob...