7.2 For the situation of Example 7.3 with m = n:

(a) Show that a necessary condition for (7.5) to converge to N(0, 1) is that √n(λˆ −λ) →

0, where λˆ = σˆ 2/τˆ2 and λ = σ2/τ 2, for σˆ 2 and τˆ2 of (7.4).

(b) Use the fact that λ/λ ˆ has an F-distribution to show that √n(λˆ − λ) → 0.

(c) Show that the full MLE is given by the solution to

ξ = (m/σ2)X¯ + (n/τ 2)Y¯

m/σ2 + n/τ 2 , σ2 = 1 m(Xi − ξ )

2

, τ 2 = 1 n

(Yj − ξ )

2

, and deduce its asymptotic efficiency from Theorem 5.1.