Sequential Cross-Validated Bandwidth Selection Under Dependence and   Anscombe-Type Extensions to Random Time Horizons
  To detect changes in the mean of a time series, one may use previsible detection procedures based on nonparametric kernel prediction smoothers which 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, e.g., 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 \cite{KurtzProtter1996}'s results on the weak convergence of \ito 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 continuous between those discontinuities.   In applications, the time horizon where monitoring stops latest is often determined by a random experiment, e.g. a first-exit stopping time applied to a cumulated cost process or a risk measure, possibly stochastically dependent from the monitored time series. Thus, we also study that case and establish related limit theorems in the spirit of \citet{Anscombe1952}'s result. The result has various applications including statistical parameter estimation and monitoring financial investment strategies with risk-controlled early termination, which are briefly discussed. 
Ansgar Steland
math.PR math.ST stat.AP stat.TH