Exponential confidence region based on the projection density estimate. Recursivity of these estimations

TL;DR

This paper explores modifications to Tchentzov's projection density estimation to create a recursive form, maintaining convergence speed while improving practical applicability for statistical tail analysis.

Contribution

It introduces a recursive version of projection density estimators assuming square integrability, enhancing usability without sacrificing convergence speed.

Findings

Recursive estimators retain classical convergence rates.

Modified estimators are more convenient for applications.

Exponential tail bounds are derived for the estimators.

Abstract

We investigate the famous Tchentzov's projection density statistical estimation in order to deduce the exponential decreasing tail of distribution for the natural normalized deviation. We modify these estimations assuming the square integrability of estimated function, to make it recursive form, which is more convenient for applications, however they have at the same speed of convergence as the for the classical ones in the composite Hilbert space norm.