(i) Under the assumptions made at the beginning of Section 5.6, the UMP unbiased test of H : ρ = ρ0 is given by (5.44).

(ii) Let (ρ, ρ¯) be the associated most accurate unbiased confidence intervals for

ρ = aγ + bδ, where ρ = ρ

(a, b), ρ¯ = ¯ρ

(a, b). Then if f1 and f2 are increasing functions, the expected value of f1(| ¯ρ − ρ|) + f2(|ρ − ρ|) is an increasing function of a2/n + b2.

[(i): Make an orthogonal transformation from y1,..., yn to new variables z1,...,zn, such thatz1 =

i[bvi + (a/n)]yi /



(a2/n) + b2,z2 =

i(avi − b)yi /

√a2 + nb2, and apply Problems 5.5 and 5.6.

(ii): If a2 1 /n + b2 1 < a2 2 /n + b2 2, the random variable | ¯ρ(a2, b2) − ρ| is stochastically larger than | ¯ρ(a1, b1) − ρ|, and analogously for ρ.]