If X1,...,Xn and Y1,...,Yn are samples from N(, 2) and N(, 2) respectively, the problem of

If X1,...,Xn and Y1,...,Yn are samples from N(ξ, σ2) and N(η, τ 2) respectively, the problem of testing τ 2 = σ2 against the two-sided alternatives τ 2 = σ2 remains invariant under the group G generated by the transformations X

i = aXi +

b, Y

i = aYi +

c, (a = 0), and X

i = Yi, Y

i = Xi.

There exists a UMP invariant test under G with rejection region W = max (Yi − Y¯ )

2

(Xi = X¯)

,

(Xi = X¯)

2

(Yi − Y¯ )2



≥ k.

[The ratio of the probability densities of W for τ 2/σ2 = ∆ and τ 2/σ2 = 1 is proportional to [(1 + w)/(∆ + w)]n−1 + [(1 + w)/(1 + ∆w)]n−1 for w ≥ 1. The derivative of this expression is ≥ 0 for all ∆.]