Question

3.10 Let , X, be 1.1.d. r.v.'s with mean and variance Ơ2, both unknown. Then for any known constants c, , c., consider the linear estimate of μ defined by: (i) Identify the condition that the G's must satisfy, so that u' is an unbiased estimate of . (ii) Show that the sample mean X is the unbiased linear estimate of u with the smallest variance 1-1 (among all unbiased linear estimates of H). Hint. For part (ii), one has to minimize the expression subject to the side restriction that f For this minimization, use the Lagrange multipliers method, which calls for the minimization of the (cı , . . .,c")-Y2 + λ1y_ function ノwith respect to c, , cn,where λ is a constant (Lagrange multiplier). Alternatively, one may employ a Geometric argument to the same effect.

Answer 1