非时齐Poisson过程MLE的重对数律(英文)

The LIL of the MLE of the Nonhomogenous Poisson Processes

在时间区间[0,T]上,我们观察到X（T）={X(t,θ)0≤t≤T},其中θ∈Θ为参数,且知.Kutoyants讨论了非时齐过程参数θ的最大似然估计（MLE）的性质,他给出了极限分布,并且得到了弱收敛及矩收敛等结果,但他要求参数空间Θ为有限区间(α,β)。本文讨论了非时齐Poisson过程MLE的性质,我们允许参数空间随时间而不断增大,即θ_T是θ在Θ_T=(α,β_T)上的最大似然估计,其中 limβ_T=∞,在一定条件下证明了θ_T满足重对数律。

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 unknown to the observer, is observed. It is required to construct sufficiently good estimates θT = θT (-XT) of the parameter 9 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. θT is the MLE on T= (α, βT), where limT-T- =oo. We will give some conditions, under which, the LIL of the MLB θT holds.