Best average power.

(i) Consider the general linear hypothesis H in the canonical form given by

(7.2) and (7.3) of Section 7.1, and for any ηr+1,...,ηs, σ, and ρ let S =

S(ηr+1,...,ηs, σ : ρ) denote the sphere {(η1,...,ηr) :

r i=1 η2 i /σ2 = ρ2}. If

βφ(η1,...,ηr,σ) denotes the power of a test φ of H, then the test (7.9) maximizes the average power

#

S βφ(η1,...,ηr,σ) d A

#

S d A for every ηr+1,...,ηs, σ, and ρ among all unbiased (or similar) tests. Here d A denotes the differential of area on the surface of the sphere.

(ii) The result (i) provides an alternative proof of the fact that the test (7.9) is UMP among all tests whose power function depends only on r i=1 η2 i /σ2.

[(i): if U = r i=1 Y 2 i , V = n i=s+1 Y 2 i , unbiasedness (or similarity) implies that the conditional probability of rejection given Yr+1,..., Ys, and U + V equals α a.e.

Hence for any given ηr+1,...,ηs, σ, and ρ, the average power is maximized by rejecting when the ratio of the average density to the density under H is larger than a suitable constant C(yr+1,..., ys, u + v), and hence when g(y1,..., yr; η1,...,ηr) = 
S exp
r i=1 ηi yi σ2 
d A > C(yr+1,..., ys, u + v).
As will be indicated below, the function g depends on y1,..., yr only through u and is an increasing function of u. Since under the hypothesis U/(U + V)is independent of Yr+1,..., Ys and U + V, it follows that the test is given by (7.9). The exponent in the integral defining g can be written as r i=1 ηi yi /σ2 = (ρ√u cos β)/σ, where β is the angle (0 ≤ β ≤ π ) between (η1,...,ηr) and (y1,..., yr). Because of the symmetry of the sphere, this is unchanged if β is replaced by the angle γ between (η1,...,ηr) and an arbitrary fixed vector. This shows that g depends on the y’s only through u: for fixed η1,...,ηr, σ denote it by h(u). Let S be the subset of S in which 0 ≤ γ ≤ π/2. Then h(u) = 
S 
exp!ρ
√u cos γ
σ
"
+ exp!−ρ
√u cos γ
σ
" d A, which proves the desired result.]