Suppose that x y 0 and consider the estimation of sysxh

Question

Suppose that X and Y are two samples with p.d.f. given by (4.30).

(a) Suppose that µ_x = µ_y = 0 and consider the estimation of η = (σ_y/σ_x)^h with a fixed h ≠ 0 under the loss L(a/η). Show that the problem is invariant under the transformations g(X, Y ) = (rX, r′Y ), r > 0, r′ > 0. Generalize.

(b) Generalize the result in (a) to the case of unknown µx and µy under the transformations in (4.31).