The reproducing kernel Hilbert space approach in nonparametric regression problems with correlated observations

TL;DR

This paper introduces a new nonparametric regression estimator for correlated time series data using reproducing kernel Hilbert spaces, providing asymptotic analysis and empirical comparison with classical methods.

Contribution

It proposes a novel estimator based on the inverse autocovariance matrix and analyzes its bias, variance, and IMSE, comparing it to the classical Gasser and Muller estimator.

Findings

The new estimator has favorable asymptotic properties.

Simulation results show improved finite-sample performance.

Theoretical analysis confirms the estimator's efficiency.

Abstract

In this paper we investigate the problem of estimating the regression function in models with correlated observations. The data is obtained from several experimental units each of them forms a time series. We propose a new estimator based on the inverse of the autocovariance matrix of the observations, assumed known and invertible. Using the properties of the Reproducing Kernel Hilbert spaces, we give the asymptotic expressions of its bias and its variance. In addition, we give a theoretical comparison, by calculating the IMSE, between this new estimator and the classical one proposed by Gasser and Muller. Finally, we conduct a simulation study to investigate the performance of the proposed estimator and to compare it to the Gasser and Muller's estimator in a finite sample set.