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

(i) If βˆj denotes the values of the βj minimizing (Xi − ξi)2 and if θˆ = s j=1 e jβˆj = n j=1 di Xi , the rejection region of the hypothesis H : θ = θ0 is


Xi − ξˆ
i 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 θˆ − k $%%&
Xi − ξˆ
i 2 n − s ≤ θ ≤ θˆ + k $%%&
Xi − ξˆ
i 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 , then ξ = β1a1 +···+ βsas, and since ξˆ is in  also ξˆ = βˆ
1a1 +···+ βˆ
sas. The space ω 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 ω. By (7.1), the vector X is given by X = n i=1 Yi ci , and its projection ξˆ on  therefore satisfies ξˆ = s i=1 Yi ci . Equating the two expressions for ξˆ and taking the inner product of both sides of this equation with ci gives Y1 = βˆ
1 n i=1 ai1ci1, since the c’s are an orthogonal set of unit vectors.
This shows that Y1 is proportional to βˆ
1 and, since the variance of Y1 is the same as that of the X’s, that |Y1|=|βˆ
1|/
 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.]