Let {X1, X2, ..., Xn} be a random sample from some population, such that E[X1] = 0. We know that a good estimator of the population mean is the sample mean X̄n. But suppose that you construct the following unusual estimator of E[X1]:
μ̂n = 0 with probability 1 - 1/n
= n with probability 1/n
(a) Find the bias of μ̂n.
(b) Find the variance of μ̂n.
(c) Calculate the MSE of μ̂n and show how it behaves as n → ∞.
(d) Intuitively, is μ̂n a consistent estimator of E[X1]? Explain your answer in the context of your response to (c).