Question

Consider a large population which has population mean µ, and population variance σ 2 . We take a sample of size n = 3 from this population, thinking of the samples as realizations of the RVs X1, X2, and X3, where the Xi can be considered iid. We are interested in estimating µ.

(a) Consider the estimator ˆµ1 = X1 + X2 − X3. Is this estimator unbiased for µ? Explain your answer.

(b) Find the variance of ˆµ1.

(c) When estimating µ, would you prefer the estimator ˆµ1 or the estimator ˆµ = X¯? Explain your answer.

(d) Now consider the estimator ˆµ2 = (X1 + X2 + X3)/2. Is this estimator unbiased for µ? Explain your answer.

(e) Compute the MSE of ˆµ2.

(f) When estimating µ, would you prefer the estimator ˆµ2 or the estimator ˆµ = X¯? Explain your answer.

Answer 1

(a) We have the three estimators X1, X2 and X3, each of which has a population mean .

Consider the expectation of the given estimator;

.

Therefore, the estimator is unbiased.

(b) To find the variance of the estimator, note that the 3 samples are considered to be independent and identically distributed with population variance . Therefore,

, since all the covariance terms are equal to 0 since the samples are independent.

(c) We know that the estimator is unbiased for estimating and has a variance , since we have 3 samples here. Note that, this variance is lower than the variance of , thereby, suggesting that is a better estimator.

(d) Note that;

which is not same as . Therefore, the given estimator is not unbiased.