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Unformatted text preview: using g (u) = g ∗ (u), where ∞ g ∗ (u) = E [Y |X = u] = vfY |X (v |u)dv. (4.27) −∞ We write E [Y |X ] for g ∗ (X ). The minimum MSE is M SE = E [(Y − E [Y |X ])2 ] ∞ ∞ −∞ ∞ −∞ ∞ −∞ 2 −∞ = = (v − g ∗ (u))2 fY |X (v |u)dv fX (u)du (4.28) v 2 − (g ∗ (u))2 fY |X (v |u)dv fX (u)du (4.29) = E [Y ] − E [(E [Y |X ])2 ]. (4.30) In summary, the minimum MSE unconstrained estimator of Y given X is E [Y |X ] = g ∗ (X ) where g ∗ (u) = E [Y |X = u], and expressions for the MSE are given by (4.28)-(4.30). 4.10.3 Linear estimators In practice it is not always possible to compute g ∗ (u). Either the integral in (4.27) may not have a closed form solution, or the conditional density fY |X (v |u) may not be available or might be difficult to compute. The problems might be more than computational. There might not even be a good way to decide what joint pdf fX,Y to use in the first place. A reasonable alternative to using g ∗ is to consider linear estimators of Y gi...
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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 Institute of Technology.

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