Let denote the sample average from a random sample with mean μ and variance σ2. Consider...

Let denote the sample average from a random sample with mean μ and variance σ2. Consider two alternative estimators of μ: W1 = [(n-1)/n] and W2= /2.

(i) Show that W1 and W2 are both biased estimators of μ and find the biases. What happens to the biases as n →∞? Comment on any important differences in bias for the two estimators as the sample size gets large.

(ii) Find the probability limits of W1 and W2. {Hint: Use Properties PLIM.1 and PLIM.2; for W1, note that plim [(n-1)/n] = 1.} Which estimator is consistent?

(iii) Find Var(W1) and Var(W2).

(iv) Argue that W1 is a better estimator than if μ is gcloseh to zero. (Consider both bias and variance.)