Concentrated Likelihood Function
We often encounter in practice the situation where the parameter vector 0O can be naturally partitioned into two subvectors Oq and fi0 as 0O = (a£, fi’0)’. The regression model is one such example; in this model the parameters consist of the regression coefficients and the error variance. First, partition the maximum likelihood estimator as 0 = (o’, fi’)’. Then, the limit distribution of •Jr{& — Oo) can easily be derived from statement (4.2.23). Partition the inverse of the asymptotic variancecovariance matrix of statement (4.2.23) conformably with the partition of the parameter vector as
(4.2.39)
Then, by Theorem 13 of Appendix 1, we have v’TXd – Oo) — N[0, (A – BC‘BT1].
Let the likelihood function be L(a, fi). Sometimes it is easier to maximize L in two steps (first, maximize it with respect to fi, insert the maximizing value of fi back into L; second, maximize L with respect to a) than to maximize L simultaneously for a and fi. More precisely, define
L*(a) = L[a, fi(a)],
where fi(a) is defined as a root of








Next, differentiating both sides of the identity
3 log L*{a) _ 3 log L[a, fla)] да да
with respect to a yields
Combining (4.2.51) and (4.2.52) yields
which implies
Finally, we have proved that (4.2.44), (4.2.49), and (4.2.54) lead precisely to the conclusion (4.2.40) as desired.
Leave a reply