Let X1,..., Xn be independently normally distributed with known variance 2 0 and means E(Xi) = i

Let X1,..., Xn be independently normally distributed with known variance σ2 0 and means E(Xi) = ξi , and consider any linear hypothesis with s ≤ n

(instead of s < n which is required when the variance is unknown). This remains invariant under a subgroup of that employed when the variance was unknown, and the UMP invariant test has rejection region



Xi − ˆ

ξˆ

i

2

−



Xi − ξˆ

i

2

= 

ξˆ

i − ˆ

ξˆ

i

2

> Cσ2 0 (7.65)

with C determined by

 ∞

C

χ2 r (y) dy = α. (7.66)

Section 7.3