Sequential Cross-Validated Bandwidth Selection Under Dependence and Anscombe-Type Extensions to Random Time Horizons

Abstract

To detect changes in the mean of a time series, one may use previsible detection procedures based on nonparametric kernel prediction smoothers that cover various classic detection statistics as special cases. Bandwidth selection, particularly in a data-adaptive way, is a serious issue and not well studied for detection problems. To ensure data adaptation, we select the bandwidth by cross-validation but in a sequential way leading to a functional estimation approach. This article provides the asymptotic theory for the method under fairly weak assumptions on the dependence structure of the error terms, which cover, for example, GARCH(p, q) processes, by establishing (sequential) functional central limit theorems for the cross-validation objective function and the associated bandwidth selector. It turns out that the proof can be based in a neat way on Kurtz and Protter's (1996) results on the weak convergence of Itô integrals and a diagonal argument. Our gradual change-point model covers multiple change-points in that it allows for a nonlinear regression function after the first change-point possibly with further jumps and Lipschitz continuity between those discontinuities. In applications, the time horizon where monitoring stops latest is often determined by a random experiment; for example, a first-exit stopping time applied to a cumulated cost process or a risk measure, possibly stochastically dependent on the monitored time series. Thus, we also study that case and establish related limit theorems in the spirit of Anscombe's (1952) result. This is achieved by embedding the stopped processes into a sequence of processes, which allows us to handle the randomly determined time horizon as a random change of time problem. The result has various applications including statistical parameter estimation and monitoring financial investment strategies with risk-controlled early termination, which are briefly discussed.