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Some nonparametric tests for change-point detection based on the \(\mathbb{P}\)-\(\mathbb{P}\) and \(\mathbb{Q}\)-\(\mathbb{Q}\) plot processes. (English) Zbl 1319.62094

Summary: We propose nonparametric procedures for testing change-point by using the \(\mathbb{P}\)-\(\mathbb{P}\) and \(\mathbb{Q}\)-\(\mathbb{Q}\) plots processes. The limiting distributions of the proposed statistics are characterized under the null hypothesis of no change and also under contiguous alternatives. We give an estimator of the change-point coefficient and obtain its strong consistency. We introduce the bootstrapped version of \(\mathbb{P}\)-\(\mathbb{P}\) and \(\mathbb{Q}\)-\(\mathbb{Q}\) processes, requiring the estimation of quantile density, and obtain their limiting laws. Finally, we propose and investigate the exchangeable bootstrap of the empirical \(\mathbb{P}\)-\(\mathbb{P}\) plot and \(\mathbb{Q}\)-\(\mathbb{Q}\) plot processes which avoids the problem of the estimation of quantile density, which is of its own interest. These results are used for calculating \(p\)-values of the proposed test statistics. Emphasis is placed on the explanation of the strong approximation methodology.

MSC:

62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference
62H15 Hypothesis testing in multivariate analysis
60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
60G09 Exchangeability for stochastic processes
60G15 Gaussian processes
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