Convergence of U-processes in Hölder spaces with application to robust detection of a changed segment

Abstract

To detect a changed segment (so called epedimic changes) in a time series, variants of the CUSUM statistic are frequently used. However, they are sensitive to outliers in the data and do not perform well for heavy tailed data, especially when short segments get a high weight in the test statistic. We will present a robust test statistic for epidemic changes based on the Wilcoxon statstic. To study their asymptotic behavior, we prove functional limit theorems for U -processes in Hölder spaces. We also study the finite sample behavior via simulations and apply the statistics to a real data example.If the changed segment is rather short compared to the sample size, tests who give higher weight to short segments have more power. Asymptotic critical values for such tests have been proved by Sigmund [23] in Gaussian case (see also [22]). The logarithmic case was treated in Kabluchko and Wang [14], whereas the regular varying case in Mikosch and Račkauskas [16]. Yao [26] and Hušková [12] compared tests with different wheightings. Račkauskas and Suquet [20], [21] have suggested to use a compromise weighting, that allows to express the limit distribution of the test statistic as a function of a Brownian motion. However, in order to apply the continuous mapping theorem for this statistic, it is necessary to establish the weak convergence of the partial sum process to a Brownian motion with respect to the Hölder norm.It is well known that the CUSUM statistic is sensitive to outliers in the data, see e.g. Prášková and Chochola [19]. The problem becomes worse if higher weights are given to shorter segments. A common strategy to obtain a robust change point test is to adapt robust two-sample test like the Wilcoxon one. This was first used by Darkhovsky [4] and by Pettitt [18] in the context of detecting at most one change in a sequence of independent observations. For a comparison of different change point test see Wolfe and Schechtmann [25]. The results on the Wilcoxon type change point statistic were generalized to long range dependent time series by Dehling, Rooch, Taqqu [6]. The Wilcoxon statistic can either be expressed as a rank statistic or as a (two-sample) U -statistic. This motivated Csörgő and Horváth [3] to study more general U -statistics for change point detection, followed by Ferger [9] and Gombay [11]. Orasch [17] and Döring [8] have studied U -statistics for detecting multiple change-points in a sequence of independent observations. Results for change point tests based on general two-sample U -statistics for short range dependent time series were given by Dehling, Fried, Garcia, Wendler [5], for long range dependent time series by Dehling, Rooch, Wendler [7].Gombay [10] has suggested to use a Wilcoxon type test also for the epidemic change problem. The aim of this paper is to generalize these results in three aspects: to study more general U -statistics, to allow the random variable to exhibit some form of short range dependence, and to introduce weightings to the statistic. This way, we obtain a robust test...

Change-point detection based on weighted two-sample U-statistics

“…Applying Chebyshev's inequality, we obtain that (19) converges to 0 in probability as n → ∞. By stationarity, this also holds for (20). An analogous procedure leads to…”

“…Zhang, Wang, and Shao [29] propose an adaptive change-point test based on U-statistics. Račkauskas and Wendler [20] apply U-statistics for the detection of epidemic changes.…”

Stein’s method of exchangeable pairs in multivariate functional approximations

“…Moreover, functional limit theorems for rescaled U-statistics have found applications in the field of changepoint analysis (see e.g. [22,23,31,32,34,35,38,52]), where it is particularly useful to know the functional limits of the related test statistics. On the other hand, the Erdős-Renyi random graph model has found numerous applications in various fields (see [13]), including epidemic modelling [1] and modelling of evolutionary conflicts [12].…”

Functional Convergence of Sequential U-processes with Size-Dependent Kernels

“…in [CH97, Gom04, RW19] such an invariance principle has been the starting point of a fruitful line of research, focussing on changepoint testing procedures based on generalisations of Wilcoxon-Mann-Whitney statistics. Further possible extensions of the results of this section, involving in particular antisymmetric kernels [CH88, GH95, CH97, Fer01] and rescaled U -processes [CH88,Fer94,RW19], are outside the scope of the present paper and will be investigated elsewhere.…”