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

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

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

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)