The Neyman‐Pearson theory for testing statistical hypotheses

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.