Statistical test
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
:
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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
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for all , where
.
References
[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 |
Statistical test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Statistical_test&oldid=12586