1 Let T1 and T2 be unbiased estimators of with varT1 2 0

1 Let T1 and T2 be unbiased estimators of , with var(T1) = 2 > 0 and var(T2) = 2 > 0, such 1 2 that 2 2. Suppose 2 = 2/2 (0, 1] is known and free of . Consider the combined 1 2 1 2 linear estimator T given by T = T1 + (1 )T2, for < < +. Note that T is not necessarily a convex combination of T1 and T2 (i.e., can be negative). Let = corr(T1, T2), with (1, 1). Note that = 1 implies that T1 = T2 with probability 1; also, = 1 is not possible since T1 and T2 are both unbiased. The following problems reinforce our intuition that an unbiased estimator T of formed as a linear combination of the available unbiased estimators T1 and T2 can be “better” (i.e., has smaller variance) than either T1 or T2, when is selected optimally (i.e., by minimizing the variance). a Show that T is also unbiased for . b Obtain var(T ), the variance of the combined estimator T . c Show that min, given below, minimizes var(T ): 1 min = . 1 2 + 2 d Evaluate var(T ) at = min and show that it is given by 2 2 varmin(T ) = var(T )|=min = 1 (1 ) . 1 2 + 2

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