(15 points). Suppose that X1 and X2 are independent random
samples from a population with mean θ, and variance σ 2 . Also, it
is the case that ˆθ1, ˆθ2, and ˆθ3 are all point estimators for a
parameter θ, the mean of our population X.
a. (5 points) Suppose further, that E[ ˆθ1] = aθ + b, where a
and b are nonzero constants. What is the bias of estimator ˆθ1?
(You can leave your answer in terms of a, b & θ.)
b. (4 points) Show that ˆθ2 = 0.25X1 + 0.75X2 and ˆθ3 = 1.25X1 −
0.25X2 are both unbiased.
c. (6 points) Find the relative efficiency of ˆθ3 with respect
to ˆθ2.