Asymptotic Properties of ML Estimators

8.4.1. Introduction

Without the conditions (c) in Definition 8.1, the solution 60 = argmax0e© E [ln(Ln (в))] may not be unique. For example, if Zj = cos(Xj + в0) with the Xj’s independent absolutely continuously distributed random variables with common density, then the density function f (z|e0) of Zj satisfies f (z|e0) = f (z|e0 + 2s n) for all integers s. Therefore, the parameter space © has to be chosen small enough to make в0 unique.

Also, the first – and second-order conditions for a maximum of E [ln(Ln (в))] at в = в0 may not be satisfied. The latter is, for example, the case for the likelihood function (8.11): if в < в0, then E[ln(Ln(в))] = —сю; if в > в0, then E[ln(Ln(в))] = —n ■ ln(в), and thus the left derivative of E[ln(Ln(в))] in в = в0 is limS;0(E[ln(Ln(в0))] — E[ln(Lnв — 5))])/5 = ю, and the right – derivative is lim5;0(E[ln(Ln(в0 + 5))] — E[ln(Ln(в0))])/5 = – п/в0. Because the first – and second-order conditions play a crucial role in deriving the asymp­totic normality and efficiency of the ML estimator (see the remainder of this section), the rest of this chapter does not apply to the case (8.11).