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Combining cumulative sum change-point detection tests for assessing the stationarity of univariate time series. (English) Zbl 1418.62305
The authors consider the problem of testing stationarity of given univariate time series data. In particular, given data $$X_1,\ldots,X_N$$, and defining the $$h$$-dimensional random vectors $$\mathbf{Y}_i^{(h)}=(X_i,\ldots,X_{i+h-1})$$, consider the null hypothesis that $$\mathbf{Y}_1^{(h)},\ldots,\mathbf{Y}_{N-h+1}^{(h)}$$ have the same distribution function. The authors propose testing this hypothesis by combining two tests for departures from stationarity: one to test for homogeneity in the univariate distribution functions of $$X_1,\ldots,X_N$$, and another to test for homogeneity in serial dependence via the copulas governing the $$\mathbf{Y}_i^{(h)}$$.
In order to achieve this aim, the authors first propose a general procedure for combining two (or more) dependent bootstrap-based hypothesis tests, investigating validity and consistency of this procedure. A test for serial dependence, based on empirical copulas, is proposed, and the limiting distribution under the null hypothesis is given. Since this distribution is intractable, the authors consider the computation of approximate $$p$$-values. Homogeneity in the univariate distribution functions is investigated via a CUSUM test, and this is combined with the copula-based procedure to give a test for stationarity in the data.
This procedure relies on the choice of an embedding dimension $$h$$, which the authors discuss. They also give some related combined tests for second-order stationarity. Finally, the paper concludes with a simulation study investigating properties of the proposed procedure, and its illustration via two real-world data sets.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G10 Nonparametric hypothesis testing 62E20 Asymptotic distribution theory in statistics 62G09 Nonparametric statistical resampling methods 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62G20 Asymptotic properties of nonparametric inference
##### Software:
BootWPTOS; copula; copula; costat; fractal; locits; npcp; R
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