Computational limits to nonparametric estimation for ergodic processes

TL;DR

This paper proves fundamental limitations in nonparametric estimation of binary ergodic processes, showing that no countable class of estimators can universally accurately estimate certain zero-entropy processes.

Contribution

It introduces a new negative result demonstrating the impossibility of universal nonparametric estimation for all binary ergodic processes within any countable estimator class.

Findings

No countable estimator class can universally estimate all zero-entropy ergodic processes.

The result differs from previous negative results by focusing on nonparametric estimation.

It establishes fundamental limits on the accuracy of ergodic process estimation.

Abstract

A new negative result for nonparametric estimation of binary ergodic processes is shown. I The problem of estimation of distribution with any degree of accuracy is studied. Then it is shown that for any countable class of estimators there is a zero-entropy binary ergodic process that is inconsistent with the class of estimators. Our result is different from other negative results for universal forecasting scheme of ergodic processes.