Marshall’s lemma for convex density estimation

Dümbgen, Lutz, University of Bern; Rufibach, Kaspar, University of Bern; Wellner, Jon A., University of Washington

doi:10.1214/074921707000000292

VOL. 55 | 2007 Marshall’s lemma for convex density estimation

Chapter Author(s) Lutz Dümbgen, Kaspar Rufibach, Jon A. Wellner

Editor(s) Eric A. Cator, Geurt Jongbloed, Cor Kraaikamp, Hendrik P. Lopuhaä, Jon A. Wellner

IMS Lecture Notes Monogr. Ser., 2007: 101-107 (2007) DOI: 10.1214/074921707000000292

Abstract

Marshall's lemma is an analytical result which implies $\sqrt{n}$--consistency of the distribution function corresponding to the Grenander estimator of a non-decreasing probability density. The present paper derives analogous results for the setting of convex densities on $[0,\infty)$.