Question

5. Consider a simple case with only four independently and identically distributed (iid) observations, X1, X2, X3, X4, on a random variable X. Consider these two estimators:

µˆ1 = 1/12 (2X1 + 4X2 + 4X3 + 2X4), µˆ2 = 1/12 (X1 + 5X2 + 5X3 + X4).

a Show that each is unbiased, and that one is more efficient than the other.

b Show that the usual sample mean is more efficient than either. Explain why the others given above are less efficient. (Optional: prove that in iid samples, for any weighted estimator ˆµa = Pn i=1 aiXi , with Pn i=1 ai = 1, then the most efficient estimator is that in which ai = 1 n for all i, ie the sample mean.)

c If the variance of each Xi is 2, how many (independent) observations do you need in order to get a standard error of .03 or less with the sample mean estimator?

Answer 1