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Random Walks with Drift – A Sequential Approach

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  • Ansgar Steland

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 associated sequential partial sum process under non‐standard sampling. The asymptotic behaviour differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results, we consider local nonparametric 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. Furthermore, we provide a theoretical result about the optimal kernel for a given alternative.

Suggested Citation

  • Ansgar Steland, 2005. "Random Walks with Drift – A Sequential Approach," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(6), pages 917-942, November.
    • Handle: RePEc:bla:jtsera:v:26:y:2005:i:6:p:917-942
    DOI: 10.1111/j.1467-9892.2005.00450.x
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    References listed on IDEAS

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    1. Steland, Ansgar, 2006. "A bootstrap view on dickey-fuller control charts for AR(1) series," Technical Reports 2006,01, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.

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