Statistical test

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A decision rule according to which a decision is taken in the problem of statistical hypotheses testing (cf. Statistical hypotheses, verification of) on the basis of results of observations.

Assume that the hypothesis : has to be tested against the alternative : by means of the realization of a random vector that takes values in a sample space , . Furthermore, let be an arbitrary -measurable function, mapping the sample space onto the interval . In a case like this, the principle according to which is rejected with probability , while the alternative is rejected with probability , is called a statistical test for testing against ; is the critical function of the test. The function , , is called the power function of the test.

The use of a statistical test leads either to a correct decision being taken, or to one of the following two errors being made: rejection of , and thus acceptance of , when in fact is correct (an error of the first kind), or acceptance of when in fact is correct (an error of the second kind). One of the basic problems in the classical theory of statistical hypotheses testing is the construction of a test that, given a definite upper bound , , for the probability of an error of the first kind, would minimize the probability of an error of the second kind. The number is called the significance level of the statistical test.

In practice, the most important are non-randomized statistical tests, i.e. those with as critical function the indicator function of a certain -measurable set in :

Thus, a non-randomized statistical test rejects if the event takes place; on the other hand, if the event takes place, then is accepted. The set is called the critical region of the statistical test.

As a rule, a non-randomized statistical test is based on a certain statistic , which is called the test statistic, and the critical region of this same test is usually defined using relations of the form , , . The constants , , called the critical values of the test statistic , are defined from the condition ; in these circumstances one speaks in the first two cases of one-sided statistical tests, and in the third case, of a two-sided statistical test. The structure of reflects the particular nature of the competing hypotheses and . In the case where the family possesses a sufficient statistic , it is natural to look for the test statistic in the class of sufficient statistics, since

for all , where .


[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2] J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)
[3] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[4] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[5] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[6] M.S. Nikulin, "A result of Bol'shev's from the theory of the statistical testing of hypotheses" J. Soviet Math. , 44 : 3 (1989) pp. 522–529 Zap. Nauchn. Sem. Mat. Inst. Steklov. , 153 (1986) pp. 129–137
How to Cite This Entry:
Statistical test. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098