65. Tukey :s T-Method. Let X; (i = 1, .. . , r) be independent N(L I), and consider simultaneous confidence intervals (116) L[(i,j) ; x) s ~j - t s M[(i ,j); x) for all i *" j . The problem of determining such confidence intervals remains invariant under the group Go of all permutations of the X's and under the group G2 of translations gx = x +

a. (i) In analogy with (61), attention can be restricted to confidence bounds satisfying (117) L[(i,j);x) = -M[(j,i);x). (ii) The only simultaneous confidence intervals satisfying (117) and equivariant under Go and G2 are those of the form (118) S( x) = {~: xj - x, - £\ < ~j - < xj - x, + £\ for all i *" j} . (iii) The constant £\ for which (118) has probability y is determined by (119) po{maxIAj - X;I < £\} = po{ XCn ) - Xci) < £\} = v, where the probability Po is calculated under the assumption that ~I = ... =t·