对Pitman估计量理论的贡献
Contribution to the theory of Pitman estimators
摘要：
本文证明了由总体F（x-θ）的固定大小样本构造θ的Pitman估计（最小方差同变估计）的方差的新的不等式。这些不等式与经典的Fisher信息的Stam不等式密切相关，它在小样本中的模拟，以及一个强大的方差下降不等式。唯一需要的条件是F的有限方差;甚至不假设F的绝对连续性。作为小样本的主要不等式的推论，人们获得了Fisher信息的已知性质的替代证明，以及有趣的新观察结果，例如基于大小为n的样本的Pitman估计量的方差在n中单调减少。推广的结果多项式版本的Pitman估计和多元位置参数。此外，搜索的不等式之一的等式条件的表征导致柯西型函数方程的独立随机变量，和一个有趣的新的行为，其解决方案进行了说明。
英文摘要：
New inequalities are proved for the variance of the Pitman estimators (minimum variance equivariant estimators) of θconstructed from samples of fixed size from populations F(x-θ). The inequalities are closely related to the classical Stam inequality for the Fisher information, its analog in small samples, and a powerful variance drop inequality. The only condition required is finite variance of F; even the absolute continuity of F is not assumed. As corollaries of the main inequalities for small samples, one obtains alternate proofs of known properties of the Fisher information, as well as interesting new observations like the fact that the variance of the Pitman estimator based on a sample of size n scaled by n monotonically decreases in n. Extensions of the results to the polynomial versions of the Pitman estimators and a multivariate location parameter are given. Also, the search for characterization of equality conditions for one of the inequalities leads to a Cauchy-type functional equation for independent random variables, and an interesting new behavior of its solutions is described.
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