Rojo, Javier; Batún-Cutz, José On the estimation of symmetric distributions under peakedness order constraints. (English) Zbl 1271.62076 Rojo, Javier (ed.), Optimality. The third Erich L. Lehmann symposium. Selected papers based on the presentations at the symposium, Rice University, Houston, TX, USA, May 16–19, 2007. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-77-5/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 57, 147-172 (2009). Summary: Consider distribution functions \(F\) and \(G\) and suppose that \(F\) is more peaked about \(a\) than \(G\) is about \(b\). The problem of estimating \(F\) or \(G\), or both, when \(F\) and \(G\) are symmetric, arises quite naturally in applications. The empirical distribution functions \(F_n\) and \(G_m\) will not necessarily satisfy the order constraint imposed by the experimental conditions. In a previous paper, the authors proposed some estimators that are strongly uniformly consistent when both \(m\) and \(n\) tend to infinity. However the estimators fail to be consistent when only either \(m\) or \(n\) tend to infinity. Here estimators are proposed that circumvent these problems and the asymptotic distribution of the estimators is delineated. A simulation study compares these estimators in terms of mean squared error and bias behavior with their competitors.For the entire collection see [Zbl 1203.62003]. Cited in 1 Document MSC: 62G05 Nonparametric estimation 60F05 Central limit and other weak theorems 62E15 Exact distribution theory in statistics 62G30 Order statistics; empirical distribution functions Keywords:partial orders; nonparametric statistics; stochastic order; empirical processes; weak convergence PDFBibTeX XMLCite \textit{J. Rojo} and \textit{J. Batún-Cutz}, IMS Lect. Notes, Monogr. Ser. 57, 147--172 (2009; Zbl 1271.62076)