Nonparametric estimation of a trend based upon sampled continuous processes

TL;DR

This paper develops nonparametric methods for estimating the trend function of a continuous stochastic process from multiple sampled realizations, providing asymptotic normality results and confidence bands.

Contribution

It introduces a novel nonparametric estimator for the trend function of a stochastic process with asymptotic normality and confidence band derivations.

Findings

Asymptotic normality of the estimator established

Simultaneous confidence bands derived using Gaussian process theory

Estimator performs well as sample size and sampling points increase

Abstract

Let X be a second order random process indexed by a compact interval [0,T]. Assume that n independent realizations of X are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function mu = EX in the space C([0,T]) as n, p go to infinity and, using Gaussian process theory, we derive approximate simultaneous confidence bands for mu.