In Example 14.4.5, consider the case of a composite null hypothesis with 0 given by (14.80). Show that the null distribution of the likelihood ratio statistic given by (14.81) is χ2 p. Hint: First consider the case a = 0, so that 0 is a linear subspace of dimension k − p. Let Z = −1/2X, so 2 log(Rn) = inf

θ∈0

|Z − −1/2

θ|

2 .

As θ varies in 0, −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.