# 182 which is a circle centered at the origin with

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online