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# 4112 key properties of the bivariate normal

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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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