In the estimation problem of the mean function of an inhomogeneous Poisson process there is a class of kernel type estimators that are asymptotically efficient alongside with the empirical mean function. We start by describing such a class of estimators which we call first order efficient estimators. To choose the best one among them we prove a lower bound that compares the second order term of the mean integrated square error of all estimators. The proof is carried out under the assumption on the mean function Λ(·) that Λ(τ) = S, where S is a known positive number. In the end, we discuss the possibility of the construction of an estimator which attains this lower bound, thus, is asymptotically second order efficient.

中文翻译：

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关于泊松过程的二阶估计的下界：渐近效率

在非均匀泊松过程的均值函数的估计问题中，存在一类核类型估计器，它们与经验均值函数一起渐近有效。我们从描述这样的估计器开始，我们称其为一阶有效估计器。为了选择其中最好的一个，我们证明了一个下限，该下限比较了所有估计量的均值平方误差的二阶项。证明是在假设平均函数Λ（·）的前提下进行的，其中Λ（τ）= S，其中S是已知的正数。最后，我们讨论了构造一个达到此下限的估计器的可能性，因此，它是渐近二阶有效的。