Robust Wilcoxon‐Type Estimation of Change‐Point Location Under Short‐Range Dependence

Abstract

We introduce a robust estimator of the location parameter for the change-point in the mean based on Wilcoxon statistic and establish its consistency for L 1 near-epoch dependent processes. It is shown that the consistency rate depends on the magnitude of the change. A simulation study is performed to evaluate the finite sample properties of the Wilcoxon-type estimator under Gaussianity as well as under heavy-tailed distributions and disturbances by outliers, and to compare it with a CUSUM-type estimator. It shows that the Wilcoxon-type estimator is equivalent to the CUSUM-type estimator under Gaussianity but outperforms it in the presence of heavy tails or outliers in the data.

Robust discrimination between long‐range dependence and a change in mean

“…Notice that a linear process or AR process might not be absolutely regular, but it would be L 1 near epoch dependent; see Example 2.1 in (Gerstenberger, 2018) for linear processes and (Hansen, 1991) for GARCH(1,1) processes. More examples of L 1 NED processes can be found in (Borovkova et al 2001), who also discuss more general L r NED processes, r ≥ 1.…”

“…An important step of our testing procedure is the estimation of the location k * of the change-point in mean. Gerstenberger (2018) showed that under Assumptions 1 and 2 the Wilcoxon-type change-point location estimator…”

“…where C > 0 is a constant. Hence, y) is a degenerate kernel, that is, E g n (x, Y 1 ) = E g n (Y 1 , y) = 0. h n (x, y) satisfies (26) and (27) with h n ( ) = C , see for example, Corollary 4.1 of Gerstenberger (2018), where constant C does not depend on n. Then, with similar argument as in Remark 5.1, h 1,n and h 2,n are 1-continuous and therefore, g n (x, y) is 1-continuous with function g n ( ) = C satisfying (32). Hence, g n (x, y) satisfies the conditions on g(x, y) in Lemma 5.4, which yields…”

“…In both articles the simulation studies point out that the Wilcoxon test statistic (2) is more robust to outliers than the CUSUM statistic (1). Recently, Gerstenberger (2018) showed that Wilcoxon-type change-point location estimator for a change in mean of short-range dependent data based on test statistic (2) is also robust against outliers.…”

Consistency of an Estimator for Change Point in Volatility of Financial Returns

“…An extensive literature on change point detection and estimation in independent and identically distributed random variables is by Csorgo and Horváth (1997) where mostly a change in mean is considered, or in martingale difference sequences as in Bai (1994) or even change in variance under the assumption that financial returns are independent and identically distributed random variables Inclan and Tiao (1994) which is not a fact when one is working with real financial data. It has over time been done in time series like Chen et al (2005), Ross (2013), Aminikhanghahi and Cook (2017) under different assumptions on dependence, heterogeneity among others Gerstenberger (2018).…”