Nonparametric Statistics

Abstract

In light of Cohen (Ann Math Stat 37:458–463, 1966) and Rao (Ann Stat 4:1023–1037, 1976), who provide necessary and sufficient conditions for admissibility of linear smoothers, one realizes that many of the well-known linear nonparametric regression smoothers are inadmissible because either the smoothing matrix is asymmetric or the spectrum of the smoothing matrix lies outside the unit interval [0, 1]. The question answered in this chapter is how can an inadmissible smoother transformed into an admissible one? Specifically, this contribution investigates the spectrum of various matrix symmetrization schemes for k-nearest neighbor-type smoothers. This is not an easy task, as the spectrum of many traditional symmetrization schemes fails to lie in the unit interval. The contribution of this study is to present a symmetrization scheme for smoothing matrices that make the associated estimator admissible. For k-nearest neighbor smoothers, the result of the transformation has a natural interpretation in terms of graph theory. P.-A. Cornillon University of Rennes, IRMAR UMR 6625, Rennes, France e-mail: pac@univ-rennes2.fr A. Gribinski Department of Mathematics, Princeton University, Princeton, NJ, USA e-mail: aurelien.gribinski@princeton.edu N. Hengartner Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: nickh@lanl.gov T. Kerdreux UMR 8548, Ecole Normale Supérieure, Paris, France e-mail: thomas.kerdreux@inria.fr E. Matzner-Løber ( ) CREST, UMR 9194, Cepe-Ensae, Palaiseau, France e-mail: eml@ensae.fr © Springer Nature Switzerland AG 2018 P. Bertail et al. (eds.), Nonparametric Statistics, Springer Proceedings in Mathematics & Statistics 250, https://doi.org/10.1007/978-3-319-96941-1_1 1 2 P.-A. Cornillon et al.