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Unformatted text preview: of the ﬁve constants in the formula (4.33) for L∗ (X ) are the means and standard deviations.
If the random variables X and Y are standard (i.e. have mean zero and variance one), then
L∗ (X ) = ρX,Y X and the MSE is 1 − ρ2 . This fact highlights the central role of the correlation
coeﬃcient, ρX,Y , in the linear estimation of Y from X.
This section covers three types of estimators of Y : unconstrained estimators of the form g (X ),
linear estimators of the form L(X ) = aX + b, and constant estimators, of the form δ. Of the
three, the unconstrained estimators are the most general, a subset of them consists of the linear
estimators, and a subset of the linear estimators is the set of constant estimators. The larger the
class of estimators optimized over, the smaller the resulting MSE is. Thus, the following ordering
among the three MSEs holds:
E [(Y − g ∗ (X ))2 ]
MSE for g ∗ (X )=E [Y |X ] 2
≤ σY (1 − ρ2 ) ≤
X,Y MSE for L∗ (X ) Var(Y )
MSE for δ ∗ =E [Y ]. (4.36) 4.10. MINIMUM MEAN SQUARE ERROR...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
- Spring '08
- The Land