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B1000
Title: Nonparametric density estimation with directional data under rotational symmetry Authors:  Eduardo Garcia-Portugues - Carlos III University of Madrid (Spain) [presenting]
Christophe Ley - University of Luxembourg (Luxembourg)
Thomas Verdebout - Universite Libre de Bruxelles (Belgium)
Abstract: Rotational symmetry is a recurrent methodological assumption in directional statistics. Most of the classical distributions for directional or axial data are rotationally symmetric (Fisher-von Mises-Langevin, Watson, Wrapped Cauchy, etc) and the roots of recently developed statistical methods rely on this property. Rotational symmetry is exploited for constructing a constrained semiparametric Kernel Density Estimator (KDE). The estimator is obtained by means of a new operator, termed \textit{rotasymmetrizer}: applied to a KDE, it ensures that the resulting estimator, the RKDE, is rotationally symmetric. The operator is based on the tangent-normal decomposition and connects the RKDE with an adapted KDE in the domain $[-1,1]$. The main properties of the RKDE are derived (bias, variance, asymptotic normality, error measurement), being the most relevant the variance order $(nh)^{-1}$ for arbitrary dimension. These properties hold with the axis of rotational symmetry either known or estimated $\sqrt{n}$-consistently. The improvement in performance with respect to the KDE is checked empirically in a simulation study for fixed and data-driven bandwidths. Finally, some applications of the RKDE in testing are discussed.