摘要:
点过程是一个应用广泛的统计模型.在医学,社会学,经济学,电子与通信科学以及软件与硬件可靠性等许多科学领域都能找到应用点过程的例子.在这些实际应用中,一般是根据问题的实际背景假定模型具有一定的参数形式,然后根据观测数据给出未知参数的极大似然估计值以推断事物发展的客观规律.我们知道,一种估计量是否收敛以及收敛速度的快慢,是决定这种估计量好坏的最为重要的标准.本文对于一般的点过程模型中向量参数极大似然估计(MLE)首先给出了一个保证其强相合的较为广泛的充分性条件,然后在进一步的条件下得到了重对数型的收敛速度。
Abstract:
Suppase {N(t), t≥0} is a point process on probability space (Ω,F,Pθ) with intensity λ(θ,t), where θ=(θ1,…’,θm) is unknown parameter. Let θ*=(θ1*,… θm*)' be the true value of θ. The observed data are{N(t), 0 ≤t≤T}. Denote the MLE of θ as ^θT. In this paper we give that, under some conditions, the MLE of θ is consistent, and $\begin{aligned} & \limsup _{T \rightarrow \infty} \frac{\lambda_{\min }(T)\left\|\widehat{\theta}_T-\theta^*\right\|}{\sqrt{2 h^2(T) \mathrm{LLg}\left(h^2(T)\right)}} \stackrel{\text { a.s. }}{\leq} 1 \\ & \limsup _{T \rightarrow \infty} \frac{c \lambda_{\max }(T)\left\|\widehat{\theta}_T-\theta^*\right\|}{\sqrt{2 h^2(T) \mathrm{LLg}\left(h^2(T)\right)}} \stackrel{\text { a.s. }}{\geq} 1\end{aligned}$ where c > 0 and h(T) are obtained from λ(t,θ*), λmax(T)(λmin(T)) is the maximum (minimum) among theeigenvalues of the information matrix.