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\%< Q < 50\%) } $ for the statistics $ R _ {n} ^ {+} { ( a, 1) } $ and $ R _ {n} { ( a, 1) } $:

$$ \sqrt { { \frac{ 1-a }{ na } } } \Phi ^ {-1} { { ( 1-0} .005 Q) } \ \textrm{ and } \ \ \sqrt { { \frac{ 1-a }{ na } } } L ^ {-1} { { ( 1-0} .01 Q) } , $$

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\% < Q < 10\% $, 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)