Statistical Decision Functions

作者：Abraham Wald · 发表于：The Annals of Mathematical Statistics · 年份：1949 · DOI：10.1214/aoms/1177730030 · 被引用次数：2203 · 研究领域：Decision-Making and Behavioral Economics、Bayesian Modeling and Causal Inference、Advanced Statistical Process Monitoring

The foundations of a general theory of statistical decision functions, including the classical non-sequential case as well as the sequential case, was discussed by the author in a previous publication [3]. Several assumptions made in [3] appear, however, to be unnecessarily restrictive (see conditions 1-7, pp. 297 in [3]). These assumptions, moreover, are not always fulfilled for statistical problems in their conventional form. In this paper the main results of [3], as well as several new results, are obtained from a considerably weaker set of conditions which are fulfilled for most of the statistical problems treated in the literature. It seemed necessary to abandon most of the methods of proofs used in [3] (particularly those in section 4 of [3]) and to develop the theory from the beginning. To make the present paper self-contained, the basic definitions already given in [3] are briefly restated in section 2.1. In [3] it is postulated (see Condition 3, p. 207) that the space $\Omega$ of all admissible distribution functions $F$ is compact. In problems where the distribution function $F$ is known except for the values of a finite number of parameters, i.e., where $\Omega$ is a parametric class of distribution functions, the compactness condition will usually not be fulfilled if no restrictions are imposed on the possible values of the parameters. For example, if $\Omega$ is the class of all univariate normal distributions with unit variance, $\Omega$ is not compact. It is tr...