As shown below x and y are the means of x and y

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Unformatted text preview: the constant δ ∗ that minimizes the mean square error for estimation of a random variable Y by a constant is the mean, and the minimum possible value of the mean square error for estimating Y by a constant is Var(Y ). Another way to derive this result is to use the fact that E [Y − EY ] = 0 and EY − δ is constant, to get E [(Y − δ )2 ] = E [((Y − EY ) + (EY − δ ))2 ] = E [(Y − EY )2 + 2(Y − EY )(EY − δ ) + (EY − δ )2 ] = Var(Y ) + (EY − δ )2 . From this expression it is easy to see that the mean square error is minimized with respect to δ if and only if δ = EY , and the minimum possible value is Var(Y ). 4.10.2 Unconstrained estimators Suppose instead that we wish to estimate Y based on an observation X. If we use the estimator g (X ) for some function g, the resulting mean square error (MSE) is E [(Y − g (X ))2 ]. We want to find g to minimize the MSE. The resulting estimator g ∗ (X ) is called the unconstrained optimal estimator of Y based on X because no constrain...
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