This is a more general version of Problem C.1. Let Y₁, Y₂,c, Y_n be n pairwise uncorrelated random variables with common mean m and common variance s2. Let Y denote the sample average.

(i) Define the class of linear estimators of µ by

where the α_i are constants. What restriction on the ai is needed for Wa to be an unbiased estimator of µ?

(ii) Find Var (W_a).

For any numbers α₁, α₂, . . ., α_n, the following inequality holds:

Use this, along with parts (i) and (ii), to show that Var(W_a) ≥ Var(Y) whenever W_a is unbiased, so that Y is the best linear unbiased estimator.

Transcribed Image Text:

Wa = a,Y, + a,Y, + . + a,Y