Suppose that x1 and x2 are random samples of observations from a population with mean m and variance s2. Consider the following three point estimators, X, Y, Z, of m:

X =

1 2

x1 +

1 2

x2 Y =

1 4

x1 +

3 4

x2 Z =

1 3

x1 +

2 3

x2

a. Show that all three estimators are unbiased.

b. Which of the estimators is the most efficient?

c. Find the relative efficiency of X with respect to each of the other two estimators.