Consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden Markov models

TL;DR

This paper develops uniform error bounds for filtering and smoothing in nonparametric hidden Markov models with unknown parameters, extending spectral estimation methods and providing explicit convergence rates.

Contribution

It introduces explicit uniform error bounds for filtering and smoothing in nonparametric HMMs and extends spectral estimation techniques to this setting.

Findings

Uniform bounds for filtering and smoothing errors in total variation norm.

Explicit L2-risk bounds for spectral estimators in nonparametric HMMs.

Simulation results demonstrating the effectiveness of spectral methods.

Abstract

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the L2-risk of the nonparametric spectral estimators. When the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions.