Nonparametric inference for ergodic, stationary time series

TL;DR

This paper presents a simple, transparent algorithm for strongly consistent nonparametric estimation of the conditional distribution in stationary, ergodic time series, extending previous finite and Polish space cases.

Contribution

It introduces a new, straightforward method for nonparametric inference in ergodic, stationary time series, applicable beyond finite or Polish spaces.

Findings

Algorithm is simple and transparent

Provides strong consistency in estimation

Extensions to regression and forecasting are discussed

Abstract

The setting is a stationary, ergodic time series. The challenge is to construct a sequence of functions, each based on only finite segments of the past, which together provide a strongly consistent estimator for the conditional probability of the next observation, given the infinite past. Ornstein gave such a construction for the case that the values are from a finite set, and recently Algoet extended the scheme to time series with coordinates in a Polish space. The present study relates a different solution to the challenge. The algorithm is simple and its verification is fairly transparent. Some extensions to regression, pattern recognition, and on-line forecasting are mentioned.