Density estimation from an individual numerical sequence

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).

💡 Research Summary

The paper tackles the problem of estimating a univariate probability density from a single deterministic numerical sequence, abandoning the usual stochastic assumptions such as independent and identically distributed (i.i.d.) samples. Instead, the authors introduce two deterministic conditions that the sequence must satisfy.

Condition (i) – limiting relative frequencies – requires that for every Borel set A the empirical proportion of terms falling in A converges to the integral of an unknown density f over A. Formally,
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