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Gold standard process Markovian poisoning: a semiparametric approach

Claire Lacour, Pierre Vandekerkhove

TL;DR

The paper develops a semiparametric framework for chronological mixtures that combine a gold standard stationary process with an unknown i.i.d. poisoning sequence through a latent two-state selector. It introduces two minimum-contrast estimators to recover the latent Markov transition and the nonparametric poisoning distribution, proving strong consistency and, under mixing, $\sqrt{n}$-consistency with a functional central limit theorem for the poisoning distribution. A plug-in nonparametric estimator for the poisoning cdf $F^1$ is derived and shown to converge to a Gaussian process, with explicit rates for both the parametric and nonparametric components. Theoretical results rely on identifiability via a linear independence condition and mild mixing assumptions, and numerical experiments illustrate performance under varying observability and dependence structures. The work also discusses extensions such as training-data based refinements and decoding strategies, highlighting potential cyber-security applications where mixed Markovian and i.i.d. signals occur.

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.

Gold standard process Markovian poisoning: a semiparametric approach

TL;DR

The paper develops a semiparametric framework for chronological mixtures that combine a gold standard stationary process with an unknown i.i.d. poisoning sequence through a latent two-state selector. It introduces two minimum-contrast estimators to recover the latent Markov transition and the nonparametric poisoning distribution, proving strong consistency and, under mixing, -consistency with a functional central limit theorem for the poisoning distribution. A plug-in nonparametric estimator for the poisoning cdf is derived and shown to converge to a Gaussian process, with explicit rates for both the parametric and nonparametric components. Theoretical results rely on identifiability via a linear independence condition and mild mixing assumptions, and numerical experiments illustrate performance under varying observability and dependence structures. The work also discusses extensions such as training-data based refinements and decoding strategies, highlighting potential cyber-security applications where mixed Markovian and i.i.d. signals occur.

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 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.
Paper Structure (26 sections, 11 theorems, 155 equations, 16 figures, 2 tables)

This paper contains 26 sections, 11 theorems, 155 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Chronological mixtures of stochastic processes
  3. Hidden Markov Mixtures of Markov Models (H4M)
  4. Gold standard process poisoning
  5. Identifiability and parametric estimation
  6. Asymptotic Normality
  7. Nonparametric estimation
  8. Numerical performances
  9. Two strong observability cases
  10. Challenging cases with lower observability
  11. Functional estimator behavior
  12. Concluding remarks
  13. Proofs
  14. Proof of Proposition \ref{['contrast']}
  15. Consistency proofs
  16. ...and 11 more sections

Key Result

Proposition 1

The $(\theta^*,\theta)$-deviation quantity defined in (deviation_quant) can be expressed as follows where $c_1$ and $c_2$ are only depending on $\theta$ and $\theta^*$, see (c1c2) for close form expressions. Moreover, we have the implication

Figures (16)

  • Figure 1: Illustration of a Markovian mixture of a Markov process and an i.i.d. sequence. Here $X_1=0, X_2=1,X_3=1,X_4=0$.
  • Figure 2: $\hbox{(S0)}_{strong}$ trajectory
  • Figure 3: $\hbox{(S0)}_{weak}$ trajectory
  • Figure 4: $N$-sample of $\sqrt{n}$-centered $\hat{\theta}_n$ estimators under model $\hbox{(S0)}_{strong}$, with $N=10,000$ repetitions and $n=1,000$, $3,000$ and $5,000$ observations.
  • Figure 5: (S1) trajectory
  • ...and 11 more figures

Theorems & Definitions (17)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • proof
  • Proposition 3
  • ...and 7 more