An extremal property of the conditional expectation
条件期望的一个极值性质
This note demonstrates that for bivariate random variables (X, Y) the correlation between Y and functions of X is maximized by the function f(X) = E(Y | X). The result provides a finite sample justification for the common scoring prooedure of giving the jth largest observation the score of the expected value of the jth order statistic of a standard distribution. Asymptotically the result also indicates that when one is using instrumental variables as an aid to estimation,…