Towards Model Selection for Local Log-Density Estimation. Fisher and Wilks-type theorems

Abstract

The aim of this research is to make a step towards providing a tool for model selection for log-density estimation. The author revisits the procedure for local log-density estimation suggested by Clive Loader (1996) and extends the theoretical results to finite-sample framework with the help of machinery of Spokoiny (2012). The results include bias expression from "deterministic" counterpart and Fisher and Wilks-type theorems from "stochastic". We elaborate on bandwidth trade-off $ h(n) = \arg\min O(h^p) + O_p(1/\sqrt{nh^d}) $ with explicit constants at big O notation. Explicit expressions involve (i) true density function and (ii) model that is selected (dimension, bandwidth, kernel and basis, e.g. polynomial). Existing asymptotic properties directly follow from our results. From the expressions obtained it is possible to control "the curse of dimension" both from the side of log-density smoothness and the inner space dimension.