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 variance-covariance matrix of statement (4.2.23) con­formably with the partition of the parameter vector as

image260(4.2.39)

Подпись: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

image262

(4.2.42)

 

(4.2.43)

 

We call L*(a) the concentrated likelihoodjunction of a. In this subsection we shall pose and answer affirmatively the question: If we treat L*(a) as if it were a proper likelihood function and obtain the limit distribution by (4.2.23), do we get the same result as (4.2.40)?

From Theorem 4.1.3 and its proof we have

 

image263
image264

where в+ lies between [o£, ІкріоУ] and в0. But we have

 

image265

Next, differentiating both sides of the identity

Подпись: (4.2.50)3 log L*{a) _ 3 log L[a, fla)] да да

image267 image268

with respect to a yields

image269 image270

Combining (4.2.51) and (4.2.52) yields

image271 Подпись: (4.2.54)

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.

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