Random walks - a sequential approach

TL;DR

This paper develops sequential monitoring schemes using kernel smoothing to detect nonparametric drifts in random walks, providing asymptotic analysis and optimal kernel selection under local alternatives.

Contribution

It introduces a novel sequential detection method for nonparametric drifts in random walks and analyzes its asymptotic behavior under various local alternatives.

Findings

Asymptotics of the Nadaraya-Watson estimator under non-standard sampling.

Behavior of the monitoring procedure varies with the drift's convergence rate.

Identification of the optimal kernel for specific alternatives.

Abstract

In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative.