基于正则收敛意义下决策函数空间的紧致性

The Compactness of Decision-making Functional Spaces Based on the Sense of Regular Convergence

就决策函数空间的特性进行讨论,指出任何统计决策问题总是与一个随机过程有关的,所有统计决策函数所构成的集合可以看作是一个拓扑空间.在给出了决策函数的概念、正则收敛意义下收敛的定义以及权函数的概念之后,引入了决策空间的内在度量.进而运用现代泛函分析的网格理论,就离散和连续两种情况证明了决策函数空间的紧致性.为统计决策函数在现代经济分析、预测、控制等领域的应用,从理论上奠定了基础.

In this paper, the characteristic of decision-making functional spaces is discussed. All statistical decision-making problems are related to a series of random courses. The set made up of all statistical decision-making functions can be treated as a topology space. After the definition of decision-making functions, convergence on the sense of regular convergence and weight functions are presented, the intrinsic measurement in decision-making spaces are introduced. Then the mesh theory in modem functional analysis is used to prove the compactness of decision-making functional spaces in both discrete condition and continuous condition, which establishes the bases in theory to utilize the statistical decision-making functions in the fields of modem economic analysis, pre-estimation, and control, etc.