A Class of Local Linear Estimators with Functional Data

Abstract

We introduce a local linear nonparametric estimation for the generalized regression function of a scalar response variable given a random variable taking values in a semi metric space. We establish a rate of uniform consistency for the proposed estimators. Then, based on a real data set we illustrate the performance of a particular studied estimator with respect to other known estimators

Rate of the Almost Sure Convergence of a Generalized Regression Estimate Based on Truncated and Functional Data

“…Note that all the sums containing G −1 n (Y i ) are taken for i such that G n (Y i ) = 0 . Following [1] and [7], the local linear estimator of m φ in the case of truncated data is obtained as the solution for a of the following minimization problem…”

“…Hypotheses (H1)-(H5) are standard in the nonparametric functional regression setting. The rest of the hypotheses have already been used in the literature, we refer for (H6) and (H7) to [1] and for (H8) to [7].…”

“…Remark 4.1. These hypothesis are the uniform version of the assumed conditions in the pointwise case and have already been used in the literature (see [7]). Proof.…”

“…Proof. By considering the same decompositions and notations (2)-( 5), following the same steps as in the proof of Lemma 3 and using Lemma 6 (i) in [7] instead of Lemma A.1 (i) in [1], we get under assumptions (U1)-(U4) and (U6)…”

“…For example, in [9] the uniform almost-complete convergence of the local linear conditional quantile estimator was established, while in [8] the case of a generalized regression function with functional dependent data was considered. The asymptotic normality of the local linear estimator of the conditional density for functional time series data was studied in [12] and both the pointwise and the uniform almost complete convergences, of a generalized regression estimate, were investigated in [7]. All these studies were carried in the case of complete data, however in practice, one or more truncation variables may interfere with the variable of interest and prevent its observation in a complete manner.…”