On the asymptotic behaviour of general maximum likelihood estimates for the nonregular case under nonstandard conditions

Abstract

SUMMARY In this paper we generalize Huber's (1967) results to include the nonregular case. Resistant estimators which turn out to be asymptotically unbiased in a neighbourhood of the model are given for some univariate models. Their order of consistency achieves the rate of the maximum likelihood estimates. Moreover, some of the estimates are asymptotically efficient under the central model, in the sense that they have the same The consistency of the maximum likelihood estimators has been studied by several authors, such as Doob (1934) and Cramer (1946). Later Wald (1949) gave, for the regular case, a simpler proof with weaker hypotheses that do not involve second and higher derivatives of the likelihood function. Weiss & Wolfowitz (1966, 1974) considered some nonregular cases via maximum probability estimators. The classical regularity conditions for the asymptotic distribution of maximum likelihood estimates are not satisfied by nonregular families. In some cases the estimates have the same asymptotic properties as in the regular case, while in others they are not asymptotically normally distributed. The case of a translation parameter of a truncated distribution p(x, 0) = p(x - 0), where p(t) vanishes on (-oo, 0) and satisfies p(t)-cytyas t -- 0, with y a 1 and c> 0, has been consider by Woodroofe (1972, 1974), Weiss & Wolfowitz (1973), Akahira (1975a, b) and Smith (1985). Woodroofe (1972) showed that, for y > 2, the maximum likelihood estimators have the same asymptotic properties as in the regular case and that, for y = 2, they have an asymptotic normal distribution but the order of consistency is O{(n log n)2}.