The Construction of Metric Space for Bootstrap-Cross-Validation in Density Estimator

Abstract

The working field of bootstrap-cross-validation is a set of numerous collection of empirical distributions on data space X, generated by bootstrap re-samples and its estimators whose are a discrete and an absolutely continuous type distributions respectively, and bootstrap-cross-validation needs a distance between arbitrary two distributions of this set. We considered that these distributions are all square-integrable functions by setting appropriate Radon-measure μ of integral, and therefor these distributions are belong to Hilbert space L_2(X,μ). In this paper we prove the existence of this appropriate measure by the actual construction of μ specifically. And more we can define the distance between arbitrary two distributions with an independent of appropriate measure. Notwithstanding independent, the prove of existence is necessary because of that μ plays a role of the fixed common basic measure for the set of L_2(X,μ).