非齐次泊松过程中向量参数极大似然估计的重对数律

Law of Iterated Logarithms of MLE for Nonhomogeneous Poisson Processes

对于具有向量参数的非齐次泊松过程的一般模型，论证了向量参数极大似然估计每个分量的收敛速度符合重对数律。

We study the law of iterated logarithms of maximum likelihood estimate for general nonhomogeneous Poisson processes.Suppose {N(t),t≥0} is a Poisson process on probability space {Ω,F,Pθ},where θ=(θ1,…,θm) is unknown parameter.Let θ*=(θ*1,…,θ*m) be the true value of θ,θ ^(T)=(θ ^1(T),…,θ ^m(T)) be the MLE of θ and (aij)i,j=1,…,m the inverse matrix of the information matrix,then under some conditions,for any i(1≤i≤m),the following equations holdlimsupT→∞θ ^i(T)-θ*i2aii(T)LLg(1/aii(T))=1 a.s.,liminfT→∞θ ^i(T)-θ*i2aii(T)LLg(1/aii(T))=-1 a.s..