Details of Research Outputs

用一个离散统计分布来近似一个连续的统计分布 (一维或多维) 一直是统计学研究的核心内容. 显然这个离散统计分布的支撑点集必须有代表性, 故称它们为代表点集, 或简称代表点. 选择代表点 可以有不同的考虑, 本文回顾并比较 4 类近似离散统计分布: 随机样本 (独立同分布)、修改的 Monte Carlo 方法、数论方法的样本 (伪 Monte Carlo 方法) 及在最小平方误准则下的代表点集和相应的统计 分布. 其中修改的 Monte Carlo 方法是本文新提出的. 本文比较 4 类方法在密度估计和重采样的统计 推断中的表现, 其中有一类是改进的自助法. 本文对最小平方误准则下的代表点的性质和数值算法进 行了详细回顾, 并且得到一些新结果, 例如, 随机样本的最小平方误准则的统计分布、椭球等高分布代 表点的几何结构以及椭球等高分布代表点和主成分的关系.

An important issue of how to use a discrete distribution to approximate a given continuous statistical distribution has been studied in statistics. Obviously, the support points of this discrete distribution must have a good representative in a certain sense. The set of these support points are called representative points (RPs). There are different considerations for representation. This paper reviews four approaches: Monte Carlo (i.i.d.), revised Monte Carlo, number-theoretic methods, and the mean squared error criterion. The revised Monte Carlo method is new. We compare the performance of resampling by these four methods in density estimation and statistical inference, one of which is a revised bootstrap method. The paper pays more attention to the properties of MSE (mean squared error) representative points and algorithms for their generation. Some new results are obtained, for example, the distribution of MSE, the geometric pattern of RPs of elliptical distributions and relationships between the RPs and principal components. © 2020, Science China Press. All right reserved.