When does third order efficiёncy imply fourth effiёncy?

Available online

16 Dec 2015

Found in Edition

2015, Volume 6, 602-607

Format

Full-length paper

In this article third and fourth order eﬃciency are studied in the framework of translation equivariant location estimators. We assume X_1,...,X_n i.i.d. f•− θ. By recognizing that equality in a special form of the Cauchy-Schwarz inequality leads to a certain dependence of the cumulants of the maximum likelihood estimator MLE for θ, it is shown that this MLE is fourth order eﬃcient if the underlying distribution is Gumbel. Contrary to similar results which were previously published this result is not based on symmetry.