Nonparametric tests of change-points with tapered data (Q2722250)

The problem of detecting a change-point in the properties of a process has been extensively studied. The change-point problem can be formulated in two different ways: a sequential problem called ``on-line'' and an a posteriori problem called ``off-line''. The last case is considered in this paper where one has to decide between homogeneity and a change-point after observing a given set of random data. Different kinds of changes can effect a stochastic process: changes in the mean, in the covariance structure, etc. In the present work the aim is to detect changes in the spectrum of a strictly stationary time series while assuming no change in the mean. As the distributions are a priori unknown the paper restricts to a nonparametric framework and two families of nonparametric test statistics for change-point detection using estimates of the spectral measure are built.NEWLINENEWLINENEWLINEThe results on change-point detection by \textit{D. Picard} [Adv. Appl. Probab. 17, 841-867 (1985; Zbl 0585.62151)] are extended to tapered data. For this, following \textit{R. Dahlhaus} [J. Time Ser. Anal. 4, 163-175 (1983; Zbl 0552.62068); Stochastic Processes Appl. 30, No. 1, 69-83 (1988; Zbl 0655.60033); Ann. Stat. 16, No. 2, 808-841 (1988; Zbl 0662.62100); Probab. Theory Relat. Fields 85, No. 2, 147-180 (1990; Zbl 0679.62071)], the tapers are introduced and then build two families of test statistics that are related to the Kolmogorov-Smirnov test statistics. Assuming that the process is Gaussian, the asymptotic normality of a double indexed (frequency - time) process is shown. These results allow to define the distribution of the tests under the null hypothesis where no change occurs. By numerical simulation it is shown that use of an appropriate taper depending on the sample size significantly improves the detection for small sample sizes.