Asymptotic Efficiency of the ML Estimator

The ML estimation approach is a special case of the M-estimation approach discussed in Chapter 6. However, the position of the ML estimator among the M – estimators is a special one, namely, the ML estimator is, under some regularity conditions, asymptotically efficient.

To explain and prove asymptotic efficiency, let

n

в = argmax(1/n) V’ g( 1j, в) (8.43)

ве© j=1

be an M-estimator of

Подпись: (8.44)§o = argmax E[g( 11, §)],

в ев

where again 11,…, 1n is a random sample from a k-variate, absolutely con­tinuous distribution with density f (z|§0), and © c Rm is the parameter space. In Chapter 6, I have set forth conditions such that

4П(§ _ §0) ^dNm [0, A_1 BA-1], (8.45)

Подпись: AE Подпись: d 2 g( Z 1,§0) d§0d§0T image709 Подпись: (8.46)

where

and

B = E [(dg( 11, §0)/d§0T) (dg( 11, §0>/9§0>]

Подпись: (8.47)= J {dg(z, §0)^) (dg(z, §0)/d§0) f (z| §0)dz.

Rk

As will be shown below in this section, the matrix A-1 BA-1 — Й-1 ispositive semidefinite; hence, the asymptotic variance matrix of в is “larger” (or at least not smaller) than the asymptotic variance matrix Й— 1 of the ML estimator в. In other words, the ML estimator is an asymptotically efficient M-estimator.

This proposition can be motivated as follows. Under some regularity condi­tions, as in Assumption 8.1, it follows from the first-order condition for (8.44) that

f (dg(z, во)/дв, Т) f (z|0o)dz = 0. (8.48)

image712 image713

Because equality (8.48) does not depend on the value of в0 it follows that, for all в,

Because the two vectors in (8.51) have zero expectations, (8.51) also reads

Подпись: Cov (dg(Z 1,в0) dln(f(Z 1|в0)А _A  дв0 , дв0 ) ■ (8.52)

It follows now from (8.47), (8.52), and Assumption 8.3 that

Подпись: B — A -A HiПодпись: Var1 d g( Z 1,в0)/двТ

9 ln(f (Zl|вo))/дв0

Подпись: (A- Подпись: B -A Подпись: - A Й Подпись: A-1 T-r-1 Подпись: = A-1 BA-1 - Й 1.
image722

which of course is positive semidefinite, and therefore so is

Note that this argument does not hinge on the independence and absolute continuity assumptions made here. We only need that (8.45) holds for some positive definite matrices A and B and that

, p Г/<л (B – A1

V” din(L„(во))/двТ f LVV-A Й Л

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