Guillermo Henry

Kernel Density Estimation on Riemannian Manifolds: Asymptotic Results

The paper concerns the strong uniform consistency and the asymptotic distribution of the kernel density estimator of random objects on a Riemannian manifolds, proposed by Pelletier (Stat. Probab. Lett., 73(3):297–304, 2005). The estimator is illustrated via one example based on a real data.

Robust nonparametric regression on Riemannian manifolds

In this study, we introduce two families of robust kernel-based regression estimators when the regressors are random objects taking values in a Riemannian manifold. The first proposal is a local M-estimator based on kernel methods, adapted to the geometry of the manifold. For the second proposal, the weights are based on k-nearest neighbour kernel methods. Strong uniform consistent results as well as the asymptotical normality of both families are established. Finally, a Monte Carlo study is carried out to compare the performance of the robust proposed estimators with that of the classical ones, in normal and contaminated samples and a cross-validation method is discussed.

Partly linear models on Riemannian manifolds

In partly linear models, the dependence of the response y on (x T, t) is modeled through the relationship y=x T β+g(t)+ϵ, where ϵ is independent of (x T, t). We are interested in developing an estimation procedure that allows us to combine the flexibility of the partly linear models, studied by several authors, but including some variables that belong to a non-Euclidean space. The motivating application of this paper deals with the explanation of the atmospheric SO2 pollution incidents using these models when some of the predictive variables belong in a cylinder. In this paper, the estimators of β and g are constructed when the explanatory variables t take values on a Riemannian manifold and the asymptotic properties of the proposed estimators are obtained under suitable conditions. We illustrate the use of this estimation approach using an environmental data set and we explore the performance of the estimators through a simulation study.

Robust estimators in partly linear regression models on Riemannian manifolds

Under a partly linear model we study a family of robust estimates for the regression parameter and the regression function when some of the predictors take values on a Riemannian manifold. We obtain the consistency and the asymptotic normality of the proposed estimators. Simulations and an application to a real dataset show the good performance of our proposal under small samples and contamination.

A Note on the Estimation of the Hölder Constant

In this article, we develop a nonparametric estimator for the Hölder constant of a density function. We consider a simulation study to evaluate the performance of the proposal and construct smooth bootstrap confidence intervals. Also, we give a brief review over the impossibility to decide whether a density function is Hölder.