Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.2. Rao–Cram ́er Lower Bound and Efficiency 373

y

dl( (1))

dl( (0))

(1) (0)

Figure 6.2.1: Beginning with the starting valuêθ(0), the one-step estimate is
θ̂(1), which is the intersection of the tangent line to the curvel′(θ)at̂θ(0)and the
horizontal axis. In the figure,dl(θ)=l′(θ).

The logistic distribution is similar to the normal distribution; hence, we can use

Xas our initial guess ofθ. The R functionmlelogistic, at the site listed in the

preface, computes thek-step estimates.

We close this section with a remarkable fact. The estimateθ̂(1)in equation

(6.2.32) is called theone-step estimator. As Exercise 6.2.15 shows, this estimator

has the same asymptotic distribution as the mle [i.e., (6.2.18)], provided that the

initial guessθ̂(0)is a consistent estimator ofθ. That is, the one-step estimate is an

asymptotically efficient estimate ofθ. This is also true of the other iterative steps.

EXERCISES

6.2.1.Prove thatX, the mean of a random sample of sizenfrom a distribution
that isN(θ, σ^2 ),−∞<θ<∞,is,foreveryknownσ^2 >0, an efficient estimator
ofθ.

6.2.2. Givenf(x;θ)=1/θ, 0 <x<θ, zero elsewhere, withθ>0, formally
compute the reciprocal of

nE

{[

∂logf(X:θ)

∂θ

] 2 }

.