Asymptotic Number of Roots of Cauchy Location Likelihood Equations

How many elements has Rn; how many roots are there? The standard maximum likelihood theory guarantees that one of the elements of Rn-the maximum likelihood estimate-is close to 0, and that all other elements are bounded away in probability as n -X 00 from 0. But it does not tell us how many elements there are. Let rn = card(Rn) be the number of roots. With probability one the likelihood equation has only simple roots, which are alternately local maxima and minima of the likelihood function. Hence rn is odd, there are ?/2(rn + 1) local maxima and 1/2(rn 1) local minima. Of the local maxima, one is the global maximum: the maximum likelihood estimate. Let us call the other local maxima "false maxima". These 1/2(rn 1) false maxima are an embarrassment for the maximum likelihood method of estimation; the following theorem shows that their number is really quite small.