Let Xi j (j = 1,..., mi) and Yik (k = 1,..., ni) be independently normally distributed with common variance σ2 and means E(Xi j) = ξi and E(Yi j) =

ξi + . Then the UMP invariant test of H :  = 0 is given by (7.63) with θ = , θ0 = 0 and θˆ = 
i mi ni Ni (Yi· − Xi·)

i mi ni Ni , ξˆ
i = mi j=1 Xi j + ni k=1 (Yik − θ )ˆ
Ni , where Ni = mi + ni .