Consider the problem of testing H : = in the family of densities (5.61) when it is given that > c >

Consider the problem of testing H : η = ξ in the family of densities

(5.61) when it is given that σ > c > 0 and that the point (ζ11,..., ζcNc ) of (5.62)

lies in a bounded region R containing a rectangle, where c and R are known. Then Theorem 5.11.1 is no longer applicable. However, unbiasedness of a test φ of H implies (5.66), and therefore reduces the problem to the class of permutation tests.

[Unbiasedness implies 

(φ(z)pσ,ζ (z) dz = α and hence

α = 
ψ(z)pσ,ζ (z) dz = 
ψ(z)
1 (
√2πσ)N exp − 1 2σ2 (zi j − ζi j)
2 dz for all σ > c and ζ in R. The result follows from completeness of this last family.]