On discriminating between long-range dependence and changes in mean. (English) Zbl 1112.62085

Berkes, István; Horváth, Lajos; Kokoszka, Piotr; Shao, Qi-Man

Ann. Stat. 34, No. 3, 1140-1165 (2006).

Summary: We develop a testing procedure for distinguishing between a long-range dependent time series and a weakly dependent time series with change-points in the mean. In the simplest case, under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point, and under the alternative it is long-range dependent.
We compute the CUSUM statistic \(T_n\), which allows us to construct an estimator \(\widehat{k}\) of a change-point. We then compute the statistic \(T_{n,1}\) based on the observations up to time \(\widehat{k}\) and the statistic \(T_{n,2}\) based on the observations after time \(\widehat{k}\). The statistic \(M_n= \max[T_{n,1},T_{n,2}]\) converges to a well-known distribution under the null, but diverges to infinity if the observations exhibit long-range dependence. The theory is illustrated by examples and an application to the returns of the Dow Jones index.

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10	Nonparametric hypothesis testing
65C60	Computational problems in statistics (MSC2010)

Keywords:

change-point in mean; CUSUM; long-range dependence; variance of the mean

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