Under the assumptions of Section 7.1 suppose that the means i are given by i = s

Under the assumptions of Section 7.1 suppose that the means ξi are given by

ξi = s j=1 aijβj , where the constants aij are known and the matrix A = (aij ) has full rank, and where the βj are unknown parameters. Let θ = s j=1 ejβj be a given linear combination of the βj .

(i) If βˆj denotes the values of the βj minimizing (Xi − ξi)

2 and if ˆ 

θ = s j=1 ejβˆj = n j=1 diXi, the rejection region of the hypothesis H : θ = θ0 is

|ˆθ − θ0|/

 d2 1 i



Xi − ˆξi

2

/(n − s)

> C0 , (7.63)

where the left-hand side under H has the distribution of the absolute value of Student’s t with n − s degrees of freedom.

(ii) The associated confidence intervals for θ are

ˆθ − k

;<<=

Xi − ˆξi

2 n − s ≤ θ ≤ ˆθ + k

;<<=

Xi − ˆξi

2 n − s (7.64)

with k = C0

 d2 i . These intervals are uniformly most accurate equivariant under a suitable group of transformations.

[(i): Consider first the hypothesis θ = 0, and suppose without loss of generality that θ = β1; the general case can be reduced to this by making a linear transformation in the space of the β’s. If a1,...,as denote the column vectors of the matrix A which by assumption span ΠΩ, then ξ = β1a1+···+βsas, and since ˆξ is in ΠΩ also ˆξ = βˆ1a1 + ··· + βˆsas. The space Πω defined by the hypothesis β1 = 0 is spanned by the vectors a2,...,as and also by the row vectors c2,...,cs of the matrix C of (7.1), while c1 is orthogonal to Πω. By (7.1), the vector X is given by X = n i=1 Yici, and its projection ˆξ on ΠΩ therefore satisfies ˆξ = s i=1 Yici.

Equating the two expressions for ˆξ and taking the inner product of both sides of this equation with ci gives Y1 = βˆ1

n i=1 ai1ci1, since the c’s are an orthogonal set of unit vectors. This shows that Y1 is proportional to βˆ1 and, since the variance of Y1 is the same as that of the X’s, that |Y1| = |βˆ1|/

 d2 i . The result for testing

β1 = 0 now follows from (7.12) and (7.13). The test for β1 = β0 1 is obtained by making the transformation X∗
i = Xi − aiβ0 1 .
(ii): The invariance properties of the intervals (7.64) can again be discussed without loss of generality by letting θ be the parameter β1. In the canonical form of Section 7.1, one then has E(Y1) = η1 = λβ1 with |λ| = 1/
 d2 1 while η2,...,ηs do not involve β1. The hypothesis β1 = β0 1 is therefore equivalent to η1 = η0 1, with η0 1 = λβ0 1 . This is invariant

(a) under addition of arbitrary constants to Y2 ...,Ys;

(b) under the transformations Y ∗
1 = −(Y1 − η0 1) + η0 1;

(c) under the scale changes Y ∗
i = cYi (i = 2,...,n), Y ∗
1 − η0 1 ∗ = c(Y1 − η0 1). The confidence intervals for θ = β1 are then uniformly most accurate equivariant under the group obtained from (a), (b), and

(c) by varying η0 1.]