A general framework to quantify deviations from structural assumptions in the analysis of nonstationary function-valued processes
The Annals of Statistics · 2024 · Volume 52 · Issue 2
Abstract
We present a general theory to quantify the uncertainty from imposing structural assumptions on the second-order structure of nonstationary Hilbert space-valued processes, which can be measured via functionals of time-dependent spectral density operators. The second-order dynamics are well known to be elements of the space of trace class operators, the latter is a Banach space of type 1 and of cotype 2, which makes the development of statistical inference tools more challenging. A part of our contribution is to obtain a weak invariance principle as well as concentration inequalities for (functionals of) the sequential time-varying spectral density operator. In addition, we introduce deviation measures in the nonstationary context, and derive corresponding estimators that are asymptotically pivotal. We then apply this framework and propose statistical methodology to investigate the validity of structural assumptions for nonstationary functional data, such as low-rank assumptions in the context of time-varying dynamic fPCA and principle separable component analysis, deviations from stationarity with respect to the square root distance, and deviations from zero functional canonical coherency.
Keywords
60B12, 60F17, 60J05, 62M10, 62M15, Concentration inequalities, Functional data analysis, Locally stationary processes, Martingale theory, nuclear operators, principle components, relevant hypotheses
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