%0 Journal Article %1 1985 %A Smith, Richard L. %D 1985 %I Biometrika Trust %J Biometrika %K MLE %N 1 %P pp. 67-90 %T Maximum Likelihood Estimation in a Class of Nonregular Cases %U http://www.jstor.org/stable/2336336 %V 72 %X We consider maximum likelihood estimation of the parameters of a probability density which is zero for $x < þeta$ and asymptotically α c(x - θ)α - 1 as $x þeta$ . Here θ and other parameters, which may or may not include α and c, are unknown. The classical regularity conditions for the asymptotic properties of maximum likelihood estimators are not satisfied but it is shown that, when $> 2$, the information matrix is finite and the classical asymptotic properties continue to hold. For α = 2 the maximum likelihood estimators are asymptotically efficient and normally distributed, but with a different rate of convergence. For $1 < < 2$, the maximum likelihood estimators exist in general, but are not asymptotically normal, while the question of asymptotic efficiency is still unsolved. For α ⩽ 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 θ, 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.

@article{1985, abstract = {We consider maximum likelihood estimation of the parameters of a probability density which is zero for $x < \theta$ and asymptotically α c(x - θ)α - 1 as $x \downarrow \theta$ . Here θ and other parameters, which may or may not include α and c, are unknown. The classical regularity conditions for the asymptotic properties of maximum likelihood estimators are not satisfied but it is shown that, when $\alpha > 2$, the information matrix is finite and the classical asymptotic properties continue to hold. For α = 2 the maximum likelihood estimators are asymptotically efficient and normally distributed, but with a different rate of convergence. For $1 < \alpha < 2$, the maximum likelihood estimators exist in general, but are not asymptotically normal, while the question of asymptotic efficiency is still unsolved. For α ⩽ 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 θ, 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.}, added-at = {2014-09-01T15:44:52.000+0200}, author = {Smith, Richard L.}, biburl = {https://www.bibsonomy.org/bibtex/2d34272b4642dbc701acf8a74f0081ddb/marsianus}, copyright = {Copyright © 1985 Biometrika Trust}, description = {JSTOR: Biometrika, Vol. 72, No. 1 (Apr., 1985), pp. 67-90}, interhash = {ee15a38ad501aa57557c7c93aab96ca0}, intrahash = {d34272b4642dbc701acf8a74f0081ddb}, issn = {00063444}, journal = {Biometrika}, jstor_articletype = {research-article}, jstor_formatteddate = {Apr., 1985}, keywords = {MLE}, language = {English}, number = 1, pages = {pp. 67-90}, publisher = {Biometrika Trust}, timestamp = {2014-09-01T15:44:52.000+0200}, title = {Maximum Likelihood Estimation in a Class of Nonregular Cases}, url = {http://www.jstor.org/stable/2336336}, volume = 72, year = 1985 }