幂级数分布族的一些统计性质

为了研究幂级数分布族性质,首先给出幂级数分布族的定义,并证明幂级数分布族的三个性质;其次,通过探讨常见的离散型随机变量分布在幂级数分布族下的参数结构,研究表明常见的离散型随机变量的分布均属于幂级数分布族;最后应用幂级数分布族性质确定这些离散型分布的完备充分统计量的分布形式、参数的一致最小方差无偏估计的方法以及随机变量的取值问题.得出结论:两点分布、二项分布、泊松分布、几何分布和负二项分布等常见离散型随机变量分布,在幂级数分布族概念下其函数形式与参数结构都是统一的.

Statistical properties of power series distribution

To investigate the properties of power series distribution,firstly this paper introduces the definition of power series distribution and proves its three properties.Secondly,the paper studies the parameter’s structures of the power series distribution of common discrete distributions,and finds that some common discrete distributions are power series distributions.Finally,by applying the properties,the paper determines the distribution forms of complete sufficient statistics,the uniformly minimum variance unbiased estimation and the random variable values problem.It concludes that the respective parameter’s structures and forms of common discrete distributions are uniform under the framework of power series distribution.