Asymptotic Properties of ML Estimators
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).