2 Let Y1 Y2 Yn be n pairwise uncorrelated random variables w

2. Let Y1, Y2, ...,Yn be n pairwise uncorrelated random variables with common mean and common variance 2. Let denote the sample average.

a) define the class of linear estimators of by Wa=a1Y1+a2Y2+...+anYn. Where the ai are constants. What restriction on the ai is needed for Wa to be an unbaised estimator for ?

b) Find Var(Wa)

c) For any numbers a1, a2,...,an, the following inequality holds: (a1 +a2 +...+an)2/n a12 +a22+...+an2. Use this, along with parts a and b, to show that Var(Wa)Var(Y) whenever Wa is unbiased, so that is the best linear estimator. (What does the inequality become when ai satisfy the restriction from part 1)

Solution