Abstract:
On the interval of time 0,
T, a realization of a random process
X(
T) =
X(
t), 0<
t<
T, characteristic of which, (for instance, mean) depends on a parameter
θ∈
Θ unknown to the observer, is observed. It is required to construct sufficiently good estimates
θT =
θT (
XT) of the parameter
θ from a realization of
X(
T) and describo its properties. A commonly used estimator for
θ is t e maximun likelihood estimator (MLE). The properties of MLE of non-homogenous processes have been discussed in Kutoyants (1), where he established the weak consistency, limit distributions and the convergence of the moments of the MLE. But he demanded that the parameter space was a bounded interval (
α,
β). In this paper, we will consider the non-homogenous Poisson processes. \hat\theta_T is the MLE on
ΘT= (
α,
βT), where lim
T→∞βT. We will give some conditions, under which, the LIL of the MLB \hat\theta_T holds.