Difference between revisions of "Rényi test"
Ulf Rehmann (talk | contribs) m (moved Rényi test to Renyi test: ascii title) |
Ulf Rehmann (talk | contribs) m (moved Renyi test to Rényi test over redirect: accented title) |
(No difference)
|
Revision as of 07:55, 26 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
![]() | (1) |
![]() | (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) |
Rényi test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=R%C3%A9nyi_test&oldid=23512