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A [[Statistical test|statistical test]] used for testing a simple non-parametric hypothesis  $  H _ {0} $(
 
A [[Statistical test|statistical test]] used for testing a simple non-parametric hypothesis  $  H _ {0} $(
 
cf. [[Non-parametric methods in statistics|Non-parametric methods in statistics]]), according to which independent identically-distributed random variables  $  X _ {1} \dots X _ {n} $
 
cf. [[Non-parametric methods in statistics|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) $,  
+
have a given continuous distribution function  $  F{ ( x) } $,  
 
against the alternatives:
 
against the alternatives:
  
 
$$  
 
$$  
H _ {1}  ^ {+} : \sup _ {| x | < \infty }  \psi [ F( x)] ( {\mathsf E} F _ {n} ( x) - F( x))
+
H _ {1}  ^ {+} : \sup _ {| x | < \infty }  \psi [ F{ ( x) }] { ( {\mathsf E} F _ {n} {( x) }-F{ ( x) })}
 
  >  0,
 
  >  0,
 
$$
 
$$
Line 23: Line 23:
 
$$  
 
$$  
 
H _ {1}  ^ {-} :  \inf _ {| x | <
 
H _ {1}  ^ {-} :  \inf _ {| x | <
\infty }  \psi [ F( x)]( {\mathsf E} F _ {n} ( x) - F( x)) <  0,
+
\infty }  \psi [ F{ ( x) }]{ ( {\mathsf E} F _ {n} {( x) }-F{ ( x) })<  0,
 
$$
 
$$
  
 
$$  
 
$$  
 
H _ {1} :  \sup _ {| x | < \infty } \  
 
H _ {1} :  \sup _ {| x | < \infty } \  
\psi [ F( x)] | {\mathsf E} F _ {n} ( x) - F( x) |  >  0,
+
\psi [ F{ ( x) }] | {\mathsf E} F _ {n} { { ( x) }-F{ ( x) }}  |  >  0,
 
$$
 
$$
  
where  $  F _ {n} ( x) $
+
where  $  F _ {n} { ( x) } $
 
is the empirical distribution function constructed with respect to the sample  $  X _ {1} \dots X _ {n} $
 
is the empirical distribution function constructed with respect to the sample  $  X _ {1} \dots X _ {n} $
and  $  \psi ( F  ) $,  
+
and  $  \psi { ( F  ) } $,  
 
$  \psi \geq  0 $,  
 
$  \psi \geq  0 $,  
 
is a weight function. If
 
is a weight function. If
  
 
$$  
 
$$  
\psi [ F( x)]  =  \left \{
+
\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 $
 
where  $  a $
Line 49: Line 58:
  
 
$$  
 
$$  
R _ {n}  ^ {+} ( a, 1)  = \  
+
R _ {n}  ^ {+} { ( a, 1) } = \  
\sup _ {F( x) \geq  a }   
+
\sup _ {F{ ( x) } \geq  a }   
\frac{F _ {n} ( x) - F( x) }{F(}
+
\frac{ F _ {n} { { ( x) }-F{ ( x) }}  }{ F{ ( }
  x) =
+
  x) } =
 
$$
 
$$
  
 
$$  
 
$$  
 
= \  
 
= \  
\max _ {F( X _ {(} m) ) \geq  a }   
+
\max _ {F{ ( X _ {(} m) } ) \geq  a }   
\frac{( m / n) - F( X _ {(} m) ) }{F( X _ {(} m) ) }
+
\frac{ { ( m / {n) }-F{ ( X} _ {(} m) } ) }{ F{ ( X _ {(} m) } ) }
 
  ,
 
  ,
 
$$
 
$$
  
 
$$  
 
$$  
R _ {n}  ^ {-} ( a, 1)  =  - \inf _ {F( x)
+
R _ {n}  ^ {-} { ( a, 1) } =  - \inf _ {F{ ( x) }
 
\geq  a }   
 
\geq  a }   
\frac{F _ {n} ( x) - F( x) }{F(}
+
\frac{ F _ {n} { { ( x) }-F{ ( x) }}  }{ F{ ( }
  x) =
+
  x) } =
 
$$
 
$$
  
 
$$  
 
$$  
 
= \  
 
= \  
\max _ {F( X _ {(} m) ) \geq  a }   
+
\max _ {F{ ( X _ {(} m) } ) \geq  a }   
\frac{F( X _ {(} m) ) - ( m- 1) / n }{F( X _ {(} m) ) }
+
\frac{ F{ ( X _ {(} m) }  {)- { { ( m-1) }} }  / n }{ F{ ( X _ {(} m) } ) }
 
  ,
 
  ,
 
$$
 
$$
  
 
$$  
 
$$  
R _ {n} ( a, 1)  =  \sup _ {F( x) \geq  a }   
+
R _ {n} { ( a, 1) } =  \sup _ {F{ ( x) } \geq  a }   
\frac{| F _ {n} ( x) - F( x) | }{F(}
+
\frac{ | F _ {n} { { ( x) }-F{ ( x) }}  | }{ F{ ( }
  x) =
+
  x) } =
 
$$
 
$$
  
 
$$  
 
$$  
 
= \  
 
= \  
\max \{ R _ {n}  ^ {+} ( a, 1), R _ {n}  ^ {-} ( a, 1) \} ,
+
\max \{ R _ {n}  ^ {+} { ( a, 1) }, R _ {n}  ^ {-} { ( a, 1) } \} ,
 
$$
 
$$
  
where  $  X _ {(} 1) \dots X _ {(} n) $
+
where  $  X _ { ( 1) \dots X _ { ( n) $
 
are the members of the series of order statistics
 
are the members of the series of order statistics
  
 
$$  
 
$$  
X _ {(} 1) \leq  \dots \leq  X _ {(} n) ,
+
X _ { ( 1) \leq  \dots \leq  X _ { ( n) ,
 
$$
 
$$
  
 
constructed with respect to the observations  $  X _ {1} \dots X _ {n} $.
 
constructed with respect to the observations  $  X _ {1} \dots X _ {n} $.
  
The statistics  $  R _ {n}  ^ {+} ( a, 1) $
+
The statistics  $  R _ {n}  ^ {+} { ( a, 1) } $
and  $  R _ {n}  ^ {-} ( a, 1) $
+
and  $  R _ {n}  ^ {-} { ( a, 1) } $
 
satisfy the same probability law and, if  $  0 < a \leq  1 $,  
 
satisfy the same probability law and, if  $  0 < a \leq  1 $,  
 
then
 
then
  
 
$$ \tag{1 }
 
$$ \tag{1 }
\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{ \sqrt {
+
\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{ \sqrt { {  
\frac{na}{1-}
+
\frac{ na }{ 1-a }
a } R _ {n}  ^ {+} ( a, 1) <
+
  } } R _ {n}  ^ {+} { ( a, 1) } <
 
x \right \}  = \  
 
x \right \}  = \  
2 \Phi ( x) - 1,\  x > 0,
+
2 \Phi { { ( x) }-1} ,\  x > 0,
 
$$
 
$$
  
 
$$ \tag{2 }
 
$$ \tag{2 }
\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{ \sqrt {
+
\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{ \sqrt { {  
\frac{na}{1-}
+
\frac{ na }{ 1-a }
a
+
  }
  } R _ {n} ( a, 1) < x \right \}  =  L( x),\  x > 0,
+
  } R _ {n} { ( a, 1) } < x \right \}  =  L{ ( x) },\  x > 0,
 
$$
 
$$
  
where  $  \Phi ( x) $
+
where  $  \Phi { ( x) } $
is the distribution function of the standard normal law (cf. [[Normal distribution|Normal distribution]]) and  $  L( x) $
+
is the distribution function of the standard normal law (cf. [[Normal distribution|Normal distribution]]) and  $  L{ ( x) } $
 
is the Rényi distribution function,
 
is the Rényi distribution function,
  
 
$$  
 
$$  
L( x)  =   
+
L{ ( x) } {
\frac{4} \pi  
+
\frac{ 4 } \pi  
  \sum _ { k= } 0 ^  \infty   
+
  } \sum _ {k=0} ^  \infty   
\frac{(- 1) ^ {k} }{2k+}
+
\frac{  { { (-1) }}  ^ {k} }{ 2k+1 }
\mathop{\rm exp} \left \{ -
+
  \mathop{\rm exp} \left \{ -
  
\frac{( 2k+ 1) ^ {2} \pi  ^ {2} }{8x  ^ {2} }
+
\frac{  { { ( 2k+1) }}  ^ {2} \pi  ^ {2} }{ 8x  ^ {2} }
 
  \right \} .
 
  \right \} .
 
$$
 
$$
Line 135: Line 144:
  
 
$$  
 
$$  
{\mathsf P} \{ R _ {n}  ^ {+} ( 0, 1) \geq  x \}  = \  
+
{\mathsf P} \{ R _ {n}  ^ {+} { ( 0, 1) } \geq  x \}  = \  
1 -  
+
1 - {
\frac{x}{1+}
+
\frac{ x }{ 1+x }
x ,\  x > 0.
+
  } ,\  x > 0.
 
$$
 
$$
  
 
It follows from (1) and (2) that for larger values of  $  n $
 
It follows from (1) and (2) that for larger values of  $  n $
 
the following approximate values may be used to calculate the  $  Q $-
 
the following approximate values may be used to calculate the  $  Q $-
percent critical values  $  ( 0\pct< Q < 50\pct) $
+
percent critical values  $  { ( 0\%< Q < 50\%) } $
for the statistics  $  R _ {n}  ^ {+} ( a, 1) $
+
for the statistics  $  R _ {n}  ^ {+} { ( a, 1) } $
and  $  R _ {n} ( a, 1) $:
+
and  $  R _ {n} { ( a, 1) } $:
  
 
$$  
 
$$  
\sqrt {1-
+
\sqrt {
\frac{a}{na}
+
\frac{ 1-a }{ na }
  } \Phi ^ {-} 1 ( 1 - 0.005 Q) \  \textrm{ and } \ \  
+
  } } \Phi ^ {-1} { { ( 1-0} .005 Q) } \  \textrm{ and } \ \  
\sqrt {1-
+
\sqrt {
\frac{a}{na}
+
\frac{ 1-a }{ na }
  } L ^ {-} 1 ( 1 - 0.01 Q) ,
+
  } } L ^ {-1} { { ( 1-0} .01 Q) } ,
 
$$
 
$$
  
respectively, where  $  \Phi ^ {-} 1 ( x) $
+
respectively, where  $  \Phi ^ {-1} { ( x) } $
and  $  L ^ {-} 1 ( x) $
+
and  $  L ^ {-1} { ( x) } $
are the inverse functions to  $  \Phi ( x) $
+
are the inverse functions to  $  \Phi { ( x) } $
and  $  L( x) $,  
+
and  $  L{ ( x) } $,  
respectively. This means that if  $  0\pct < Q < 10\pct $,  
+
respectively. This means that if  $  0\% < Q < 10\% $,  
then  $  \Phi ^ {-} 1 ( 1 - 0.005Q) \approx L ^ {-} 1 ( 1 - 0.02Q) $.
+
then  $  \Phi ^ {-1} { { ( 1-0} .005Q) } \approx L ^ {-1} { { ( 1-0} .02Q) } $.
  
 
Furthermore, if  $  x > 2.99 $,  
 
Furthermore, if  $  x > 2.99 $,  
Line 167: Line 176:
  
 
$$  
 
$$  
L( x)  \approx  4 \Phi ( x) - 3
+
L{ ( x) } \approx  4 \Phi { { ( x) }-3}
 
$$
 
$$
  
when calculating the values of the Rényi distribution function  $  L( x) $;  
+
when calculating the values of the Rényi distribution function  $  L{ ( x) } $;  
its degree of error does not exceed  $  5 \cdot 10 ^ {-} 7 $.
+
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
 
In addition to the Rényi test discused here, there are also similar tests, corresponding to the weight function
  
 
$$  
 
$$  
\phi [ F( x)]  =  \left \{
+
\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 $
 
where  $  a $

Latest revision as of 12:16, 8 June 2020


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)
How to Cite This Entry:
Rényi test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=R%C3%A9nyi_test&oldid=48598
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article