# 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 (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 .