In Example 12.4.5, consider the case of a composite null hypothesis with 0 given by (12.79). Show

In Example 12.4.5, consider the case of a composite null hypothesis with Ω0 given by (12.79). Show that the null distribution of the likelihood ratio statistic given by (12.80) is χ2 p. Hint: First consider the case a = 0 so that

Ω0 is a linear subspace of dimension k − p. Let Z = Σ−1/2X, so that 2 log(Rn) = inf θ∈Ω0

|Z − Σ−1/2

θ|

2 .

As θ varies in Ω0, Σ−1/2θ varies in a subspace L of dimension k − p. If P is the projection matrix onto L and I is the identity matrix, then 2 log(Rn) =

|(I − P)Z|

2.