(i) Let Xij (j = 1,...n;i = 1,...,s) be independent N(i, 2), 2 unknown. Then the problem...

(i) Let Xij (j = 1,...n;i = 1,...,s) be independent N(ξi, σ2), σ2 unknown. Then the problem of obtaining simultaneous confidence intervals for all differences ξj − ξi is invariant under G

0, G2, and the scale changes G3.

(ii) The only equivariant confidence bounds based on the sufficient statistics Xi· and S2 = (Xij − Xi·)

2 and satisfying the condition corresponding to (9.104) are those given by S(x) = 

x : xj· − xi· − ∆

√n − s S ≤ ξj − ξi (9.109)

≤ xj· − xi· +

∆

√n − s S for all i = j



with ∆
determined by the null distribution of the Studentized range P0

max |Xj· − Xi·|

S/√n − s < ∆



= γ. (9.110)

(iii) Extend the results of