Nonparametric Tests of Change‐Points with Tapered Data

Abstract

In this work we build two families of nonparametric tests using tapered data for the off-line detection of change-points in the spectral characteristics of a stationary Gaussian process. This is done using the Kolmogorov±Smirnov statistics based on integrated tapered periodograms. Convergence is obtained under the null hypothesis by means of a double indexed (frequency±time) process together with some extensions of Dirichlet and Fejer kernels. Consistency is proved using these statistics under the alternative. Then, using numerical simulations, we observe that the use of tapered data signi®cantly improves the properties of the test, especially in the case of small samples. Giratis et al. (1996) for linear processes and Lavielle (1993) for multidimensional Gaussian processes.On the other hand, it is well known that, for estimating the spectral measure of a stationary process, the use of the periodogram requires a large number of data. To bypass this problem of sample sizes, Dahlhaus has shown, in a series of papers (Dahlhaus, 1983(Dahlhaus, , 1988a(Dahlhaus, , 1988b(Dahlhaus, , 1990, that the use of tapered data improves spectral estimation. The increase in the asymptotic variance is balanced by a reduction of the bias, which leads to better results for small sample sizes. This is the old remedy to reduce leakage effects pointed out by Tukey (1967) or more recently in the papers of Zhang (1991), von Sachs (1994 and Janas and von Sachs (1995).In this paper, we extend the results on change-point detection (Picard, 1985) to tapered data. For this, following Dahlhaus, we introduce tapers, and then build two families of test statistics that are related to the Kolmogorov±Smirnov test statistics. Assuming that the process is Gaussian, we show the asymptotic normality of a double indexed (frequency±time) process. This results allows us to de®ne the distribution of our tests under the null hypothesis where no change occurs. By numerical simulation, we show that use of an appropriate taper depending on the sample size signi®cantly improves the detection for small sample size.Section 2 contains the assumptions, notation and the de®nition of the auxiliary process Z T on which the statistical studies rely. We prove that the test statistics derived from Z T converge under the null hypothesis to a known distribution that only depends on the chosen taper. Therefore, it is possible to tabulate this limiting distribution according to the taper, and obtain in practice the associated tests. The result relies on a central limit theorem for Z T stated in Theorem 2. Section 3 contains some preliminary results on the kernels used here. The functional central limit theorem for Z T is proved in Section 4. Section 5 is devoted to the study of the asymptotic distributions of our statistics under the null hypothesis. The main point is to show that the two statistics have the same asymptotic distribution (Theorem 3). In Section 6, we study the asymptotic con®dence region (Corollary 1) and the asymptotic consistency of the tests. Finally, w...

The heterogeneous Great Moderation

“…Lavielle (1993) constructs a sequential likelihood-ratio test for detecting changes in the spectrum of a multidimensional process. Picard (1985) and Rozenholc (2001) use a generalization of the Massey (1951) test for a break point in the spectrum. These are all non-parametric tests very much related to the Kolmogorov-Smirnov test.…”

Variance Change Point Detection Under a Smoothly-Changing Mean Trend with Application to Liver Procurement

“…Later a method for locating multiple change points in a Binomial Sequence was proposed in Halpern (2000). Rozenholc (2001)…”

Robust Test for Detecting Changes in the Autocovariance Function of a Time Series

“…A test based on the auto-copula has been proposed in Bücher, Fermanian, and Kojadinovic (2019). Tests that check stationarity of the spectrum are presented in Picard (1985), Giraitis and Leipus (1992), and Rozenholc (2001) and a wavelet periodogram is used in Nason (2013) and Cardinali and Nason (2018). There are also proposals that compare local estimates of the spectrum with a global estimation; see Von Sachs and Neumann (2000), Paparoditis (2009), Paparoditis (2010) Vetter (2011), andDette (2013).…”