Change point analysis of covariance functions: A weighted cumulative sum approach

We develop and study change point detection and estimation procedures for the covariance kernel of functional data based on the norms of a generally weighted process of partial sample estimates. It is shown under mild weak dependence and moment conditions on the data that in the absence of a change point a detector based on integrating such a process over the partial sample parameter is asymptotically distributed as the norm of a Gaussian process, which furnishes a consistent change point detection procedure. We further derive consistency and local asymptotic results for this detector in the presence of a change in the covariance function. The corresponding change point estimator based on such a process is also shown to be rate optimal for estimating an existing change point, and further is asymptotically distributed as the argument maximum of a Gaussian process under a local asymptotic framework. We study the detector and change point estimator in a small simulation study to detect changes in the covariance of functional autoregressive and generalized conditionally heteroscedastic processes, which demonstrate that the use of the weighted CUSUM statistics in this context generally improves performance over existing methods. These new statistics are demonstrated in an application to detecting changes in the volatility of high resolution intraday asset price curves derived from oil futures prices.