对称连续分布函数的最优不变估计

The Best Invariant Estimator of a Symmetric Continuous Distribution Function

该文考虑了未知对称连续分布函数的不变估计问题.连续分布函数在单调变换群下是不变的,但这个变换群不能保证对称分布函数的不变性.于是,所要研究的判决问题在单调变换群下不再是不变的.为了保证判决问题不变性,考虑一个新的变换群一单调奇变换群,它确保了所研究的判决问题的不变性.注意到对称分布函数零点的特殊性质,即,对任一对称分布函数F,均有F(0)=1/2,通过视零点为一伪观察值,得到了所有的非随机化不变估计,并在不变估计中找到了最优不变估计.

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, 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 ot 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 Fat：the zero point -F（O） =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.