Blockwise dynamic selection of smoothing parameters for kernel estimators of nonstationary densities of streaming data

Abstract

We consider nonparametric estimation of the nonstationary density of streaming data collected continuously over time. Those types of data are often available only for a limited time after being observed, for example due to storage or data privacy constraints. Therefore, analyzing them requires dynamic approaches that do not require repeated access to past data. A suitable iterative kernel estimator of nonstationary densities has been developed by Rutkowski (1989); it requires choosing important tuning parameters each time a new observation arises, for which no adequate practical selection criterion has been developed yet.We propose a data-driven dynamic selection of those parameters, which can be implemented iteratively and requires only sequential access to consecutive blocks of the most recent data. To choose the sizes of the blocks, we propose a theoretically founded approach based on properties of cross-validation criteria. We illustrate the broader potential of our ideas by suggesting a dynamic blockwise practical implementation of another kernel estimator of nonstationary densities developed by Hall et al. (2006).We apply our technique to simulated data and to light curve data from the NASA Kepler mission.