Page 445 - Special Topic Session (STS) - Volume 3

STS555 Patrice Bertail et al.
X(t) = X(TN(t)) × e −ωA(t) . (4)

The embedded chain of X, denoted X˜ = (X(Tn))n∈N := (Xn)n∈N, which is the
process on the intake instants T0, T1, ... plays a leading role in the analysis of X
and describes the exposure process immediately after each intake. It is defined
by the following stochastic recurrence equation

Xn+1 = Xn × e −ωΔTn+1 + Wn+1, n ≥ 0. (5)

Equation (5) is an autoregressive process with random coefficient. Under
additional assumptions [2] have related the continuous-time process X with
the embedded chain ˜X. Denote, µ(dx) = g(x)dx (resp. ˜µ(dx) = ˜g(x)dx). They
show that the limiting distribution µ and µ˜ are linked by the following
equation

1

in such a way that ∫ {(), } → µ(]0, µ]) when t → ∞. Let ˜FLA,M,n be
0
a robust estimator of (] − ∞, ]) as in example 1, then the robust estimator
is given by

Similarly to the Sparre-Andersen case, in the exponential inter-arrival case, we
have the expression

Notice that in that case, the (non-robust) plug-in estimator of µ([, ∞[) has
the nice expression

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