Page 442 - Special Topic Session (STS) - Volume 3

STS555 Patrice Bertail et al.

Definition 5. (GROSS-ERROR SENSIVITY) A functional T is said to be Markov-
robust iff its influence function T (b, L) is bounded on the torus T. The gross-
(1)
error sensitivity to block contamination is then defined as

These quantities may be estimated either with the true blocks (in the
atomic cases) or with the approximated ones in the general Harris recurrent
case.
It is now easy to see how it is possible to derive functional central limit theorems
for Fréchet differentiable functionals in a Markovian setting. for plug in estimators
base either on (in the regenerative case)

or (in the Harris general case)

ˆ
where Bi, i = 1, · · · , ln − 1 are pseudo-regeneration blocks and ˆnAM = ˆτAM
ˆ
ˆ
( ln) − ˆτAM(1) = is the total number of observations after the first
and before the last pseudo-regeneration times.
3. Examples
Example 1: Sample means. Suppose that X is positive recurrent with stationary
distribution µ. Let f: E → R be µ-integrable and consider the parameter µ(f) def
= Eµ[f (X)], f is a real function. Denote by B a r.v. valued in T with distribution LA
and observe that

with the notation f(b):= ∑ () (bi) for any b = (b1, . .., bL(b)) ∈ T. A classical
=1
calculation for the influence function of ratios yields

(1)
Notice that ELA [T (B, LA)] = 0.

431 | I S I W S C 2 0 1 9