B1102
Title: Uniform in number of neighbors consistency for the conditional $U$-statistics involving functional data
Authors: Amel Nezzal - Universite de Technologie de Compiegne (France) [presenting]
Salim Bouzebda - Universite de Technologie de Compiegne (France)
Abstract: $U$-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses.$U$-statistics generalise the empirical mean of a random variable $X$ to sums over every $m$-tuple of distinct observations of $X$. The class of so-called conditional $U$-statistics may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. We introduce the $k$ nearest neighborhoods estimator of the conditional $U$-statistics and establish uniform in $\mathbf{ t}$ and in the number of neighborhoods (UINN) (at some specific rate) to $m(\mathbf{ t})$ when $Y$ and covariates $X$ are functional taking value in some abstract spaces. In addition, uniform consistency is also established over $\varphi \in \mathcal{F}$ for a suitably restricted class $\mathcal{F}$ in both cases bounded and unbounded satisfying some moment conditions. The approaches in some recent papers are unified. The theoretical uniform consistency results are (or will be) key tools for many further developments in functional data analysis. The theorems allow data-driven local bandwidths for conditional estimators
