Consider the problem of testing H : = in the family of densities (5.61) when it is given that >c> 0 and that

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 Then Theorem 5.11.1 is no longer applicable.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

(zij − ζij )

2



dz for all σ>c and ζ in R. The result follows from completeness of this last family.]