Relative Regression on Riemannian Manifolds
Résumé
The considerations of this paper are restricted to random variables with values on Riemannian manifolds M, and hence we propose a geometric framework to estimate their relative regression function. Suppose we are given observations (X i , Y i) i=1•••n , where X i ∈ M and Y i ∈ IR + *. In this work we define and study a new estimator of the regression function on Riemannian Manifold M. Precisely, we use the mean squared relative error (MSRE) as a loss function to construct a nonparametric estimator of the regression operator on Riemannian Manifolds. Under some standard assumptions in Riemannian Manifolds data analysis, we establish the almost sure consistency, with rates, of the constructed estimator as well as its asymptotic normality. Then, a simulation study, on finite-sized samples, was carried out in order to show the efficiency of our estimation procedure.
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