2       This is a more general version of Problem C.1. Let Y1, Y2, …, Yn be n pairwise uncorrelated random variables with common mean m and common variance 2. Let Y¯ denote the sample

average.

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

Wa   a1Y1  a2Y2  …  anYn,

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

(ii)   Find Var(Wa).

(iii)  For any numbers a1, a2, …, an, the following inequality holds: (a1  a2  … 

an)2/n  a1   a2   …  an. Use this, along with parts (i) and (ii), to show  that

2            2                           2

¯                                                            ¯

Var(Wa) Var(Y) whenever Wa is unbiased, so that Y is the best linear unbiased esti-

mator. [Hint: What does the inequality become when the ai satisfy the restriction from part (i)?]