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Biometrika 1966 53(3-4):594-595; doi:10.1093/biomet/53.3-4.594
© 1966 by Biometrika Trust
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MISCELLANEA

An extremal property of the conditional expectation

DAVID. R. BRILLINGER

London School of Economics and Political Science

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, one should employ that function of the instrumental variables that may prove linearly related to the primary variables.


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