The clarification of the structure of the inverse problem in statistics and its applications

假设存在完全充分统计量，且允许含有真实的参数的实值函数的幂级数在具有一定收敛半径的圆上展开。进一步证明了在圆上存在UMVU估计的情况下，如果估计的无偏性对整个参数空间成立，则它是UMVU估计.例如，在指数和均匀的情况下，它示出在给定的参数空间中构造UMVU估计。

Suppose that there exists the complete sufficient statistic and the expansion of power series of a real-valued function of a real parameter is admitted in the circle with a certain convergence radius. Then it is shown to be possible to construct the UMVU estimator of the function in it. Further, in the case when there exists the UMVU estimator in the circle, if the unbiasedness of the estimator holds for the whole parameter space, then it is the UMVU estimator. For example, in exponential and uniform cases it is shown to construct the UMVU estimator in the given parameter space.