zbMATH — the first resource for mathematics

Hypothesis testing with the help of statistical tests using a procedure for rejecting decisions. (English. Russian original) Zbl 0842.62012
Russ. Acad. Sci., Dokl., Math. 49, No. 3, 507-510 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 3, 301-303 (1994).
In the classical Neyman-Pearson theory of hypothesis testing with the help of statistical tests for two competing alternative hypotheses \(H\) and \(H'\), an optimal test (a most powerful test) is constructed in such a way that the probability of mistakenly accepting the hypothesis \(H'\) when the hypothesis \(H\) is true (the probability \(p(H' |H)\) of an error of the first kind) does not exceed some critical level \(\alpha^*\), while the probability \(p(H |H')\) of an error of the second kind attains a minimal value. However, nothing can be said about the size \(\beta= p(H |H')\) of the probability of an error of the second kind, which can take any value from 0 to 1. This approach to hypothesis testing is not natural for many problems arising in medicine, the natural sciences, and engineering. For example, in differential diagnostics of malignant tumors (hypothesis \(H\)) from hyperplastic processes (hypothesis \(H'\)) according to given morphometric and cytospectrophotometric indicators it is necessary to construct tests so that the probabilities of errors of the first and second kind do not exceed some sufficiently small critical levels \(\alpha^*\) and \(\beta^*\): \[ p(H' |H)\leq \alpha^*, \qquad p(H|H')\leq \beta^*. \tag{1} \] If we use the Neyman-Pearson approach, then in this case we can get a most powerful statistical test for which the probability of an error of the second kind, that is, the probability of diagnosing a malignant neoplasm when there is hyperplasia of tissue, is sufficiently high, and the use of such a test in clinical practice is quite inadmissible due to the serious consequences for the patient. Unfortunately, for a fixed sample size there can fail to be a single statistical test satisfying the condition (1) if the basic hypothesis \(H\) should be accepted or rejected with certainty as a result of using the test. However, it is fairly simple to find a way out of this seemingly hopeless situation: it suffices to introduce a procedure RD for rejecting a decision, and it is possible to construct an optimal test satisfying the condition (1).
62F03 Parametric hypothesis testing
62C99 Statistical decision theory
62G10 Nonparametric hypothesis testing
62A01 Foundations and philosophical topics in statistics
91B06 Decision theory