Page 438 - Special Topic Session (STS) - Volume 3

STS555 Patrice Bertail et al.

Robust estimators for some piecewise-
deterministic markov Processes
1
2
2
Patrice Bertail ,Gabriela Ciolek ,Charles Tillier
1 Universit´e Paris Nanterre
2 TelecomParisTech

Abstract
This talk is devoted to extending the notion of robustness to Markov chains
with applications to PDMP, based on their (pseudo-) regenerative properties.
Precisely, it is shown how it is possible to define the "influence function" in this
framework, so as to measure the impact of (pseudo-) regeneration data blocks
on the statistic of interest. We establish some asymptotic results for robust
estimators of some usual functionals of the stationary measure. We also define
the concept of regeneration-based signed linear rank statistic and L-statistics,
as specific functionals of the regeneration blocks, which can be made robust
against outliers in this sense. Indeed, even the usual quantile is not robust in
this framework. We essentially apply these notions to reservoir models in
insurance and hydrology. We obtain robust estimators of probability of ruins
and quantiles (value at risk).

Keywords
Markov Chains; Piecewise Deterministic Markov processes; influence function

1. Regenerative Markov chains: Notations and context
Denote by X = (Xn)n∈N a positive recurrent Markov chain on a countably
generated state space (E, ) with transition probability Π and initial probability
ν. For any B ∈ E and n ∈ N, we have

X 0 ∼ ν and P(X n+1 ∈ B|X 0, · · ·, X n) = Π(X n, B) a.s.

In the following, Px (resp. Pν) designates the probability measure such that X0
= x and X0 ∈ E (resp. X0 ∼ ν), and Ex (·) is the Px-expectation (resp. Eν (·) is the
Pν-expectation). All along this paper, we suppose that X is ψ -irreducible and
aperiodic Markov chain. We are particularly interested in the atomic structure
of Markov chains as in [1].

Definition 1. Suppose that X is aperiodic and ψ-irreducible. We say that a set
A ∈ is an accessible atom if for all x, y ∈ A we have (x, ·) = Π(y, ·) and ψ(A) >
0. In that case we call X atomic.

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