Nonparametric regression for spherical data Marco Di Marzio Agnese Panzera DMQTE, DiSIA, Universit`adi Chieti-Pescara, Universit`adi Firenze, Viale Pindaro 42, Viale Morgagni 59, 65127 Pescara, 50134 Firenze, Italy. Italy. email:
[email protected] email:
[email protected] Charles C. Taylor Department of Statistics, University of Leeds, Leeds LS2 9JT, UK email:
[email protected] October 21, 2013 1 Authors’ Footnote: Marco Di Marzio is Associate Professor of Statistics, DMQTE, University of Chieti-Pescara, Italy. Agnese Panzera is researcher in Statistics at DISIA, University of Florence, Italy. Charles Taylor is Professor of Statistics, School of Mathematics, University of Leeds, Leeds LS2 9JT (E-mail:
[email protected]). We are grateful to an Associate Editor and two anonymous Referees for a careful reading of the manuscript and helpful comments, which led to an improved version of this article. We also thank Isabella Fabbri for having read and discussed some parts of the manuscript. 2 Abstract We develop nonparametric smoothing for regression when both the predictor and the response variables are defined on a sphere of whatever dimension. A local polynomial fitting approach is pursued, which retains all the advantages in terms of rate optimality, interpretability and ease of implementation widely observed in the standard setting. Our estimates have a multi-output nature, meaning that each coordinate is separately estimated, within a scheme of a regression with a linear response. The main properties include linearity and rotational equivariance. This research has been motivated by the fact that very few models describe this kind of regression.