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Unformatted text preview: ven X. A linear estimator has the form L(X ) = aX + b, and
to specify L we only need to ﬁnd the two constants a and b, rather than ﬁnding a whole function
g ∗ . The MSE for the linear estimator aX + b is
M SE = E [(Y − (aX + b))2 ].
Next we identify the linear estimator that minimizes the MSE. One approach is to multiply out
(Y − (aX + b))2 , take the expectation, and set the derivative with respect to a equal to zero and
the derivative with respect to b equal to zero. That would yield two equations for the unknowns a
and b. We will take a slightly diﬀerent approach, ﬁrst ﬁnding the optimal value of b as a function
of a, substituting that in, and then minimizing over a. The MSE can be written as follows:
E [((Y − aX ) − b)2 ]. 164 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES Therefore, we see that for a given value of a, the constant b should be the minimum MSE constant
estimator of Y − aX, which is given by b = E [Y − aX ] = µY − aµX . Therefore, the optimal linear
estimator has the form aX + µY − aµX , and the corresponding MSE is given by
M SE = E [(Y − a...
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- Spring '08
- The Land