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

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

This remains invariant under a subgroup of that employed when the variance was unknown, and the UMP invariant test has rejection region



Xi − ˆˆξi

2

−



Xi − ˆξi

2

=



ˆξi − ˆˆξi

2

> Cσ2 0 (7.65)

with C determined by

 ∞

C

χ2 r(y) dy = α. (7.66)

Section 7.3