Density estimation from an individual numerical sequence

TL;DR

This paper introduces a simple density estimation method for individual sequences with bounded variation, proving its consistency under certain conditions and highlighting limitations without variation bounds.

Contribution

It proposes a new density estimation scheme for individual sequences with bounded variation and establishes its $L_1$ consistency under specific assumptions.

Findings

The estimation scheme is $L_1$ consistent when the sequence's relative frequencies follow an unknown density.

No consistent estimator exists for sequences satisfying only the relative frequency condition.

The method relies on known bounds for the variation of the density over increasing intervals.

Abstract

This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that (i) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and (ii) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be $L_1$ consistent when (i) and (ii) apply. In addition it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition (i).