8. Under the assumptions of Section 1 suppose that the means t are given by s ~; = L a;/Ji, j-l where the constants aij are known and the matrix A = (aij) has full rank, and where the fJj are unknown parameters. Let 8 = Ej_,e)Jj be a given linear combination of the fJj . (i) If Pi d~notes the values of the fJj minimizing E( X; - ~;) 2 and if 0 = Et_,ejfJj = E'J_,d;X;, the rejection region of the hypothesis H: 8 = 80 is (110) 10 - 8ol/[rA YL(X; - 02/(n - s) 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 8 are (111) O-k A )2 L(X;-( ::::;8::::;0+k n-s L(X;_i;)2 n-s with k = eo/Ed;. These intervals are uniformly most accurate equivariant under a suitable group of transformations. [(i): Consider first the hypothesis 8 = 0, and suppose without loss of generality that 8 = fJ,; the general case can be reduced to this by making a linear transformation in the space of the fJ's. If g" .. . , gs denote the column vectors of the matrix A which by assumption span TID, then = fJ,g, + ... +fJsgs' and since is in TID, also = p,g, + ... +Psgs'The space II", defined by the hypothesis fJl = 0 is spanned by the vectors g2" .. , gs and also by the row vectors f2"'" fs of the matrix C of (1), while

f, is orthogonal}o TI",. By (1), the vectorAX is given by X=E7-,¥;f;, and its projection { on TID therefore satisfies { = E: _, ¥;f; . Equating the two expres-

sions for and taking the inner product of both sides of this equation with fl gives Yl ,; filE7-lQilCl;' since the f'S are an orthogonal set of unit vectors. This shows that Yl is proportional to fil and, since the variance of lJ. is the same as that of the X's, that IYll = IfitV lEd; . The result for testing PI = 0 now follows from (12) and (13). The test for PI = M is obtained by making the transformation X;* = X; - QilM· (ii): The invariance properties of the intervals (111) can again be discussed without loss of generality by letting 8 be the parameter Pl' In the canonical form of Section 1, one then has E(Yl ) = 1Jl = API with IAI = 1/lEd? while 1J2' ..., 1J, do not involvePl' The hypothesis PI = PP is therefore equivalent to 1Jl = 1J? with 1J? = APr, This is invariant

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

(b) under the transformations lJ.* = - (Yl - 1J?) + 1J?;

(c) under the scale changes Y;* = cY; (i = 2, .. . , n), yr - 1J?* = c(Yl - 1J?). The confidence intervals for 8 = PI are then uniformly most accurate equivariant under the group obtained from (a), (b), and

(c) by varying 1J?]