Gold standard process Markovian poisoning: a semiparametric approach

TL;DR

This paper introduces a semiparametric method to estimate the transition dynamics of a hidden Markov process influenced by an unknown i.i.d. poisoning sequence, providing consistent estimators and a CLT under minimal assumptions.

Contribution

It proposes two novel estimators for the transition of a hidden Markov process with an unknown distribution, demonstrating their consistency and asymptotic normality.

Findings

Both estimators are consistent for the transition parameter.

One estimator achieves $ oot n$-consistency and a functional CLT.

Numerical examples validate the estimators' performance.

Abstract

We consider in this paper a stochastic process that mixes in time, according to a nonobserved stationary Markov selection process, two separate sources of randomness: i) a stationary process which distribution is accessible (gold standard); ii) a pure i.i.d. sequence which distribution is unknown (poisoning process). In this framework we propose to estimate, with two different approaches, the transition of the hidden Markov selection process along with the distribution, not supposed to belong to any parametric family, of the unknown i.i.d. sequence, under minimal (identifiability, stationarity and dependence in time) conditions. We show that both estimators provide consistent estimations of the Euclidean transition parameter, and also prove that one of them, which is $\sqrt$ n-consistent, allows to establish a functional central limit theorem about the unknown poisoning sequence cumulative distribution function. The numerical performances of our estimators are illustrated through various challenging examples.