Let X1,...,Xm and Y1,...,Yn be samples from N(, 2) and N(, 2). The UMP unbiased test for

Let X1,...,Xm and Y1,...,Yn be samples from N(ξ, σ2)

and N(η, σ2). The UMP unbiased test for testing η − ξ = 0 can be obtained through Problems 5.5 and 5.6 by making an orthogonal transformation from (X1,...Xm, Y1,...Yn) to (Z1,...,Zm+n) such that Z1 = (Y¯ −

X¯)/

1/m + (1/n), Z2 = (Xi + Yi)/

√m + n.