Statistical Inference

Summary The subject matter of mathematical statistics may be divided into two parts, the theory of probability and the theory of inference. The first is concerned with deductions from the population to the sample; the second with inferences from the sample to the population, and may further be subdivided into the design and analysis of experiments. In the present paper we are concerned with this latter subdivision. We use the axiomatic theory of Kolmogorov as our theory of probability, and we suppose that the design of the experiment has been selected and the experiment carried out: our problem is that of analyzing the results. We suppose that the purpose of the analysis is to make decisions for future action, and that we have to choose which decision to take on the basis of the observed result of the experiment. A rule for making these decisions is called a decision function, and in §1 we show there exists a class of decision functions which is, in a sense to be explained, optimum, and in §2 we construct this class. In §3 the consequences of the previous results are discussed and we are able to appreciate what further has to be specified before a meaning can be given to the phrase “the best decision function”. In §4 these ideas are applied to some common statistical problems of classification, estimation and tests of hypotheses.