Structural break analysis for spectrum and trace of covariance operators

This paper deals with analyzing structural breaks in the covariance operator of sequentially observed functional data. For this purpose, procedures are developed to segment an observed stretch of curves into periods for which second‐order stationarity may be reasonably assumed. The proposed methods are based on measuring the fluctuations of sample eigenvalues, either individually or jointly, and traces of the sample covariance operator computed from segments of the data. To implement the tests, new limit results are introduced that deal with the large‐sample behavior of vector‐valued processes built from partial sample eigenvalue estimates. These results in turn enable the calibration of the tests to a prescribed asymptotic level. Applications to Australian annual minimum temperature curves and sea surface temperature anomaly records confirm that the proposed methods work well in finite samples. The first application suggests that the variation in annual minimum temperature underwent a structural break in the 1950s, after which typical fluctuations from the generally increasing trend started to be significantly smaller.