Let Y denote the sample average from a random sample with mean m and variance s2

. Consider two alternative estimators of m: W1 5 3 1n 2 12/n4Y and W2 5 Y/2.

(i) Show that W1 and W2 are both biased estimators of m and find the biases. What happens to the biases as n S `? 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 3 1n 2 12/n4 5 1.6 Which estimator is consistent?

(iii) Find Var1W1 2 and Var1W2 2.

(iv) Argue that W1 is a better estimator than Y if m is “close” to zero. (Consider both bias and variance.)