Page 441 - Special Topic Session (STS) - Volume 3

STS555 Patrice Bertail et al.

and

5. Drop the first block ˆB0 and the last one

2. Robust functional parameter estimation for Markov Chains
The concepts of influence function and/or robustness in the i.i.d. setting
provide tools to detect outliers among the data or influential observations.
Extending the notion of influence function and/or robustness to the general
time series framework is a difficult task; see [3] or [4]. Alternatively, the
regenerative approach offers the opportunity of extending much more
naturally an extension of the influence function based on the (approximate)
regeneration blocks construction.
The influence function on the torus
Just like the stationary probability distribution µ(dx), most parameters of
interest related to Harris positive chains are functionals of the distribution LA
of the regenerative blocks on the torus T = Un≥E , namely the distribution of
n
(X1,..., XτA) conditioned on X0 E A when the chain possesses an accessible atom
A, or the distribution of (X1,..., XτAM) conditioned on (X0, Y0) ∈ AM in the general
case when one considers the split chain. For simplicity, we shall omit the
subscript M and make no notational distinction between the regenerative and
pseudo-regenerative cases. Indeed, the probability distribution Pν of the
Markov chain X starting from ν can be factorized as follows:

where Lν means the conditional distribution of (X1, . . . , XτA) given that X0 ∼ ν. Any
functional of the law of the discrete-time process (Xn)n≥1 can be thus expressed as
a functional of the pair (Lν, LA). In the time-series asymptotic framework, since the
distribution of Lν cannot be estimated in general, only functionals of LA are of
practical interest. We propose a notion of influence function for such statistics. Let
PT denote the set of all probability measures on the torus T and for any b ∈ T, set
k
L(b) = k if b ∈ E , k > 1. We then have the following natural definition, which
straightforwardly extends the classical notion of influence function in the i.i.d. case,
with the important novelty that distributions on the torus are considered here.

Definition 4. (INFLUENCE FUNCTION ON THE TORUS) Let (V, 11.11) be a
separable Banach space. Let T : PT → V be a functional on PT. If, for all L in PT,
t (T((1 − t)L+δb) − T(L)) has a finite limit as t → 0 for any b ∈ T, the influence
−1
(1)
function T : PT → V of the functional T is then said to be well-defined, and,
by definition, one set for all b in T,
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