A tail adaptive approach for change point detection
Consider the problem of change point detection in multivariate data with unknown tails. Letting \(\mathbf{X}_1,\ldots,\mathbf{X}_n\) be \(d\)-dimensional data, which may be weakly dependent, with \(\mathbf{X}_i=(X_{i,1},\ldots,X_{i,d})\), the authors propose a tail-adaptive strategy to detect a single change point in \(E(\mathbf{X}_i)\) based on statistics of the form \[ \frac{1}{n^{3/2}}\sum_{i=1}^{\lfloor nt\rfloor}\sum_{j=\lfloor nt\rfloor +1}^n\text{sgn}(X_{j,k}-X_{i,k})|X_{j,k}-X_{i,k}|^r\,, \] where the choice of \(r\) is governed by the tails of the data. The proposed test avoids the explicit estimation of the long-run covariance matrix using a bootstrap procedure. The limiting distribution of the test statistic under the null hypothesis that \(E(\mathbf{X}_1)=\cdots=E(\mathbf{X}_n)\) is calculated in terms of a \(d\)-dimensional Brownian bridge, under certain assumptions on the structure of the data. A tail-adaptive approach, considering a range of values of \(r\) simultaneously, is justified, which overcomes the need to know the tails of the data. The paper concludes with some numerical work, including a simulation study and an application to S\&P 500 data.
scientific article; zbMATH DE number 6997743
scientific article