Bayes character and admissibility of Hotellings T 2. (i) Let (X1,...,Xp), = 1, . . .

Bayes character and admissibility of Hotelling’s T 2.

(i) Let (Xα1,...,Xαp), α = 1, . . . , n, be a sample from a p-variate normal distribution with unknown mean ξ = (ξ1,...,ξp) and covariance matrix

Σ = A−1, and with p ≤ n − 1. Then the one-sample T 2-test of H : ξ = 0 against K : ξ = 0 is a Bayes test with respect to prior distributions Λ0 and

Λ1 which generalize those of Example 6.7.13 (continued).

(ii) The test of part (i) is admissible for testing H against the alternatives

ψ2 ≤ c for any c > 0.

[If ω is the subset of points (0, Σ) of ΩH satisfying Σ−1 = A + η

η for some fixed positive definite p × p matrix A and arbitrary η = (η1,...,ηp), and Ω

A,b is the subset of points (ξ, Σ) of ΩK satisfying Σ−1 = A + η

η, ξ
= bΣη
for the same A and some fixed b > 0, let Λ0 and Λ1 have densities defined over ω and ΩA,b, respectively by

λ0(η) = C0|A + η

η|

−n/2 and

λ1(η) = C1|A + η

η|

−n/2 exp nb2 2



η(A + η

η)

−1

η





.

(Kiefer and Schwartz, 1965).]