Maximum likelihood estimation in a class of nonregular cases

SUMMARY We consider maximum likelihood estimation of the parameters of a probability density which is zero for x 2, the information matrix is finite and the classical asymptotic properties continue to hold. For cx = 2 the maximum likelihood estimators are asymptotically efficient and normally distributed, but with a different rate of convergence. For 1 < a < 2, the maximum likelihood estimators exist in general, but are not asymptotically normal, while the question of asymptotic efficiency is still unsolved. For cx < 1, the maximum likelihood estimators may not exist at all, but alternatives are proposed. All these results are already known for the case of a single unknown location parameter 0, but are here extended to the case in which there are additional unknown parameters. The paper concludes with a discussion of the applications in extreme value theory.