Rényi test

A statistical test used for testing a simple non-parametric hypothesis $ H _ {0} $( cf. Non-parametric methods in statistics), according to which independent identically-distributed random variables $ X _ {1} \dots X _ {n} $ have a given continuous distribution function $ F( x) $, against the alternatives:

$$ H _ {1} ^ {+} : \sup _ {| x | < \infty } \psi [ F( x)] ( {\mathsf E} F _ {n} ( x) - F( x)) > 0, $$

$$ H _ {1} ^ {-} : \inf _ {| x | < \infty } \psi [ F( x)]( {\mathsf E} F _ {n} ( x) - F( x)) < 0, $$

$$ H _ {1} : \sup _ {| x | < \infty } \ \psi [ F( x)] | {\mathsf E} F _ {n} ( x) - F( x) | > 0, $$

where $ F _ {n} ( x) $ is the empirical distribution function constructed with respect to the sample $ X _ {1} \dots X _ {n} $ and $ \psi ( F ) $, $ \psi \geq 0 $, is a weight function. If

$$ \psi [ F( x)] = \left \{ \begin{array}{lll} \frac{1}{F(} x) & \textrm{ when } &F( x) \geq a, \\ 0 & \textrm{ when } &F( x) < a, \\ \end{array} \right .$$

where $ a $ is any fixed number from the interval $ [ 0, 1] $, then the Rényi test, which was intended for testing $ H _ {0} $ against the alternatives $ H _ {1} ^ {+} $, $ H _ {1} ^ {-} $, $ H _ {1} $, is based on the Rényi statistics

$$ R _ {n} ^ {+} ( a, 1) = \ \sup _ {F( x) \geq a } \frac{F _ {n} ( x) - F( x) }{F(} x) = $$

$$ = \ \max _ {F( X _ {(} m) ) \geq a } \frac{( m / n) - F( X _ {(} m) ) }{F( X _ {(} m) ) } , $$

$$ R _ {n} ^ {-} ( a, 1) = - \inf _ {F( x) \geq a } \frac{F _ {n} ( x) - F( x) }{F(} x) = $$

$$ = \ \max _ {F( X _ {(} m) ) \geq a } \frac{F( X _ {(} m) ) - ( m- 1) / n }{F( X _ {(} m) ) } , $$

$$ R _ {n} ( a, 1) = \sup _ {F( x) \geq a } \frac{| F _ {n} ( x) - F( x) | }{F(} x) = $$

$$ = \ \max \{ R _ {n} ^ {+} ( a, 1), R _ {n} ^ {-} ( a, 1) \} , $$

where $ X _ {(} 1) \dots X _ {(} n) $ are the members of the series of order statistics

$$ X _ {(} 1) \leq \dots \leq X _ {(} n) , $$

constructed with respect to the observations $ X _ {1} \dots X _ {n} $.

The statistics $ R _ {n} ^ {+} ( a, 1) $ and $ R _ {n} ^ {-} ( a, 1) $ satisfy the same probability law and, if $ 0 < a \leq 1 $, then

$$ \tag{1 } \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \sqrt { \frac{na}{1-} a } R _ {n} ^ {+} ( a, 1) < x \right \} = \ 2 \Phi ( x) - 1,\ x > 0, $$

$$ \tag{2 } \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \sqrt { \frac{na}{1-} a } R _ {n} ( a, 1) < x \right \} = L( x),\ x > 0, $$

where $ \Phi ( x) $ is the distribution function of the standard normal law (cf. Normal distribution) and $ L( x) $ is the Rényi distribution function,

$$ L( x) = \frac{4} \pi \sum _ { k= } 0 ^ \infty \frac{(- 1) ^ {k} }{2k+} 1 \mathop{\rm exp} \left \{ - \frac{( 2k+ 1) ^ {2} \pi ^ {2} }{8x ^ {2} } \right \} . $$

If $ a = 0 $, then

$$ {\mathsf P} \{ R _ {n} ^ {+} ( 0, 1) \geq x \} = \ 1 - \frac{x}{1+} x ,\ x > 0. $$

It follows from (1) and (2) that for larger values of $ n $ the following approximate values may be used to calculate the $ Q $- percent critical values $ ( 0\pct< Q < 50\pct) $ for the statistics $ R _ {n} ^ {+} ( a, 1) $ and $ R _ {n} ( a, 1) $:

$$ \sqrt {1- \frac{a}{na} } \Phi ^ {-} 1 ( 1 - 0.005 Q) \ \textrm{ and } \ \ \sqrt {1- \frac{a}{na} } L ^ {-} 1 ( 1 - 0.01 Q) , $$

respectively, where $ \Phi ^ {-} 1 ( x) $ and $ L ^ {-} 1 ( x) $ are the inverse functions to $ \Phi ( x) $ and $ L( x) $, respectively. This means that if $ 0\pct < Q < 10\pct $, then $ \Phi ^ {-} 1 ( 1 - 0.005Q) \approx L ^ {-} 1 ( 1 - 0.02Q) $.

Furthermore, if $ x > 2.99 $, then it is advisable to use the approximate equation

$$ L( x) \approx 4 \Phi ( x) - 3 $$

when calculating the values of the Rényi distribution function $ L( x) $; its degree of error does not exceed $ 5 \cdot 10 ^ {-} 7 $.

In addition to the Rényi test discused here, there are also similar tests, corresponding to the weight function

$$ \phi [ F( x)] = \left \{ \begin{array}{ll} \frac{1}{1-} F( x) & \textrm{ if } F( x) \leq a, \\ 0 & \textrm{ if } F( x) > a, \\ \end{array} \right .$$

where $ a $ is any fixed number from the interval $ [ 0, 1] $.

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)