Revision as of 18:54, 24 March 2012
A statistical test used for testing a simple non-parametric hypothesis
(cf. Non-parametric methods in statistics), according to which independent identically-distributed random variables
have a given continuous distribution function
, against the alternatives:
where
is the empirical distribution function constructed with respect to the sample
and
,
, is a weight function. If
where
is any fixed number from the interval
, then the Rényi test, which was intended for testing
against the alternatives
,
,
, is based on the Rényi statistics
where
are the members of the series of order statistics
constructed with respect to the observations
.
The statistics
and
satisfy the same probability law and, if
, then
![](https://www.encyclopediaofmath.org/legacyimages/r/r081/r081270/r08127030.png) | (1) |
![](https://www.encyclopediaofmath.org/legacyimages/r/r081/r081270/r08127031.png) | (2) |
where
is the distribution function of the standard normal law (cf. Normal distribution) and
is the Rényi distribution function,
If
, then
It follows from (1) and (2) that for larger values of
the following approximate values may be used to calculate the
-percent critical values
for the statistics
and
:
respectively, where
and
are the inverse functions to
and
, respectively. This means that if
, then
.
Furthermore, if
, then it is advisable to use the approximate equation
when calculating the values of the Rényi distribution function
; its degree of error does not exceed
.
In addition to the Rényi test discused here, there are also similar tests, corresponding to the weight function
where
is any fixed number from the interval
.
References
[1] | A. Rényi, "On the theory of order statistics" Acta Math. Acad. Sci. Hungar. , 4 (1953) pp. 191–231 |
[2] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |
[3] | 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) |
How to Cite This Entry:
Rényi test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=R%C3%A9nyi_test&oldid=22973
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article