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Summary This is an attempt to write an introduction to some aspects of the present state of the Neyman‐Pearson theory. The object is to interest the reader who has some knowledge of modem probability theory and statistics. We try to emphasize the main ideas, "unnecessary" mathematics is avoided.In our opinion, the Neyman‐Pearson theory (or, more generally, the objectivistic approach to statistics) is a vigorous attempt to statisfy some weakened form of the philosophical principle of the "intersubjective verifiability". This attempt is considered successful in those situations where "various different objectivistic optimum properties are satisfied by the same optimal test". Unfortunately, these conditions are not satisfied for many situations from actual practice (testing against restricted alternatives, non‐parametric problems). Thus we arrive at a dilemma: for many problems from practice different "optimum" tests are available. There is no easy way out of this dillemma (see the last section).We have tried to emphasize the motivation and to aim at readability while minimizing the overlap with e.g. Lehmann's basic textbook. Thus many basic tools are disregarded: exponential families, completeness and sufficiency of statistics and many other subjects are missing, the restrictions by similarity, unbiasedness or invariance are only mentioned. The references constitute an incomplete apology for deleting so many important contributions.

AbstractThis paper presents some results arising out of the collaboration during 1974–1980 between an ornithologist (the first author) and a statistician. Attention is focussed on the question as to whether Tables 1 and 2 provide sufficient evidence for the existence of effects of recreation upon the numbers of oyster‐catcher nests and on the estimation of these effects by means of confidence intervals. The main tool is the standard theory for constructing conditional tests for multiparameter exponential families. The authors try to give a very careful description of the methodological aspects of their case. Many different probabilistic models can be proposed. Accordingly the statistical problems concerning tests and confidence intervals admit different mathematical formulations and (or) solutions.It is now apparent that, during their collaboration, the authors did not always make the most appropriate choice of models and procedures. They went through several stages before arriving at the choices which are basic to the sections 2, …, 5. By outlining these stages in Section 6 and mentioning the methodological errors which were committed they hope that readers will beware of similar pitfalls.

Abstract  In Part I exact results for univariate ("p= 1") two‐group ("k = 2") classification problems were derived assuming normality and equality of the variances. In Part IIa asymptotic results for multivariate ("p> I") two‐group classification and discrimination problems are based on the corresponding assumptions of multivariate normality and equality of the covariance matrices. The results (4.6.5), (4.6.6) and (4.6.7) are believed to be new.The asymptotic results in Section 4.6, together with results presented elsewhere in the literature, constitute the basis of various detailed proposals to deal with problems from actual statistical practice. Most of these proposals are modifications or specifications of existing ones. We shall pay some attention to (I) testing whether differences exist. But we are mainly interested in: (II) constructing a discriminant function, (III) assigning the individual under classification, and in (IV) constructing a confidence interval for "the" posterior probability that the individual under classification belongs to Population 2.An important part in our theory is played by various techniques for selecting variables in discriminant analysis. The need for such techniques follows from Section 4.10. The consequences of building‐in a selection technique are discussed in Section 4.12. One of our proposals motivates the theory presented in Chapter 3 and is mentioned here for that reason: employ a large part of the data, say 70%, in order to construct a discriminant function (via a selection of variables); by applying this function to the rest of the data, the exact univariate theory of Part I becomes of application. Part IIb will contain a chapter on applications.

This paper presents some of the results of the collaboration of the two authors, a mathematical‐statistician and a physical‐anthropologist, who have worked together since about 1960. Part I will consist of three chapters. Chapter I contains a description of the problems, especially concerning physical anthropology, which constitute the main motivation for this paper. Chapter 2 summarizes well‐known results which are included in order to make the paper self‐contained. Chapter 3 is mainly devoted to the univariate two‐population classification problem and based on assumptions of normality and equality of the variances. Explicit expressions (Section 3.3) and confidence intervals (Section 3.4) arc obtained for the two misclassification probabilities of the maximum likelihood rule φ* (Section 3.2) which is of application if the two kinds of error are equally bad. A Ncyman‐Pearson formulation is considered (Section 3.7). The asymptotic distribution is derived of an estimator for the posterior probability that the individual comes from Population 2 (Section 3.6). An interesting difficulty, frequently met with in physical anthropology, is that many individuals have to be classified without any (or with only small) reference samples being available from the relevant populations (Section 3.8). Part II will consist of two chapters. Chapter 4 contains new results for multivariate classification and discrimination problems. Chapter 5 contains applications.