We consider the problem of robustly detecting changepoints in the variability of a sequence of independent multivariate functions. We develop a novel changepoint procedure, called the functional Kruskal--Wallis for covariance (FKWC) changepoint procedure, based on rank statistics and multivariate functional data depth. The FKWC changepoint procedure allows the user to test for at most one changepoint (AMOC) or an epidemic period, or to estimate the number and locations of an unknown amount of changepoints in the data. We show that when the ``signal-to-noise'' ratio is bounded below, the changepoint estimates produced by the FKWC procedure attain the minimax localization rate for detecting general changes in distribution in the univariate setting (Theorem 1). We also provide the behavior of the proposed test statistics for the AMOC and epidemic setting under the null hypothesis (Theorem 2) and, as a simple consequence of our main result, these tests are consistent (Corollary 1). In simulation, we show that our method is particularly robust when compared to similar changepoint methods. We present an application of the FKWC procedure to intraday asset returns and f-MRI scans. As a by-product of Theorem 1, we provide a concentration result for integrated functional depth functions (Lemma 2), which may be of general interest.

翻译：我们考虑在独立多元函数序列的变异性中稳健检测变点的问题。基于秩统计与多元函数数据深度，我们提出一种新型变点检测方法——协方差函数Kruskal-Wallis（FKWC）变点检测程序。该程序支持用户检验至多一个变点（AMOC）、流行病期间变点，或估计数据中未知数量变点的个数及位置。我们证明：当"信噪比"有下界时，FKWC程序产生的变点估计在单变量设置下达到检测分布一般变化的极小化最优定位速率（定理1）。同时给出原假设下AMOC与流行病设置中检验统计量的渐近行为（定理2），并作为主要结果的直接推论，证明此类检验具有相合性（推论1）。仿真表明：与同类变点检测方法相比，本方法具有显著稳健性。我们将FKWC程序应用于日内资产收益与f-MRI扫描数据。作为定理1的副产品，我们给出积分函数深度函数的浓度结果（引理2），该结果可能具有普适价值。