Let Xij (j = 1,...,mi) and Yik (k = 1,...,ni) be independently normally distributed with common variance σ2 and means E(Xij ) = ξi and E(Yij ) = ξi + ∆. Then the UMP invariant test of H : ∆ = 0 is given by (7.63)

with θ = ∆, θ0 = 0 and

ˆθ =



i mini Ni (Yi· − Xi·)



i mini Ni

, ˆξi =



mi j=1 Xij + ni k=1

(Yik − ˆθ)

Ni

, where Ni = mi + ni.