Rényi test

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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


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 .


[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)
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Rényi test. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article