If X1,..., Xn is a sample from N(, 2), the UMP unbiased tests of 0

If X1,..., Xn is a sample from N(ξ, σ2), the UMP unbiased tests of

ξ ≤ 0 and ξ = 0 can be obtained from Problems 5.5 and 5.6 by making an orthogonal transformation to variables Z1,..., Zn such that Z1 = √nX¯ .

[Then n

i=2 Z2 i = n i=1 Z2 i − Z2 1 = n i=1 X2 i − nX¯ 2 = n i=1

(Xi − X¯)

2

.]