Estimation in hidden Markov models via efficient importance sampling

TL;DR

This paper introduces an importance sampling method for efficiently estimating the sampling distribution of statistics in hidden Markov models, improving bootstrap confidence region approximation.

Contribution

It develops an asymptotically optimal resampling scheme using a tilting formula derived from a LAN family, enhancing simulation efficiency in HMMs.

Findings

The proposed importance sampling scheme reduces variance in estimates.

Numerical examples demonstrate improved efficiency over traditional methods.

The tilting formula simplifies implementation via Poisson equation solutions.

Abstract

Given a sequence of observations from a discrete-time, finite-state hidden Markov model, we would like to estimate the sampling distribution of a statistic. The bootstrap method is employed to approximate the confidence regions of a multi-dimensional parameter. We propose an importance sampling formula for efficient simulation in this context. Our approach consists of constructing a locally asymptotically normal (LAN) family of probability distributions around the default resampling rule and then minimizing the asymptotic variance within the LAN family. The solution of this minimization problem characterizes the asymptotically optimal resampling scheme, which is given by a tilting formula. The implementation of the tilting formula is facilitated by solving a Poisson equation. A few numerical examples are given to demonstrate the efficiency of the proposed importance sampling scheme.