关键词: 不变估计, 对称分布函数, 非参数估计

Abstract:

This paper considers the problem of invariant estimator of an unknown symmetric continuous distribution function. Though the group of all one to one monotone transformations of real values onto themselves leaves the parametric space of all continuous distribution functions invariant^[1], it can not insure the parametric space of all the symmetric continuous distribution functions invariant. Thus, the decision problem is not invariant under the group of monotone transformations. In order to guarantee this invariance, the authors consider a new group of transformations -- the group of all the odd monotone transformations. It leaves the decision problem invariant. By noticing the special feature of a symmetric distribution function F at the zero point -F(0)=1/2 and viewing the zero point as a pseudo-observation value, the authors obtain all the nonrandomized invariant estimators and find the best invariant estimator in the invariant estimators.

Key words: Invariant estimator, Symmetrical distribution function, Non-parametric estimation