Let(Xi, Yi), i = 1 ... n be i.i.d. such that Xi and Yi are independent and normally distributed, Xi has variance σ2, Yi has variance τ 2 and both have common mean μ.

(i) If σ and τ are known, determine an efficient likelihood estimator (ELE) μˆ of μ

and find the limit distribution of n1/2(μˆ − μ).

(ii) If σ and τ are unknown, provide an estimator μ¯ for which n1/2(μ¯ − μ) has the same limit distribution as n1/2(μˆ − μ).

(iii) What can you infer from your results (i) and (ii) regarding the Information matrix I(θ), θ = (μ, σ, τ )?