非时齐Poisson过程MLE的重对数律

The LIL of the MLE of the Nonhomogenous Poisson Processes

  • 摘要: 在时间区间0,T上,我们观察到XT) = Xt,θ), 0≤tT,其中θΘ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\hat\theta_T是θΘT= (α, βT)上的最大似然估计,其中 limT→∞βT,在一定条件下证明了\hat\theta_T满足重对数律。

     

    Abstract: On the interval of time 0, T, a realization of a random process XT) = Xt), 0<tT, characteristic of which, (for instance, mean) depends on a parameterθΘ unknown to the observer, is observed. It is required to construct sufficiently good estimates θT = θTXT) of the parameter θ from a realization of XT) 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 limT→∞βT. We will give some conditions, under which, the LIL of the MLB \hat\theta_T holds.

     

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