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''non-central <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669602.png" />-distribution''
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A continuous probability distribution concentrated on the positive semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669603.png" /> with density
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{{TEX|auto}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669604.png" /></td> </tr></table>
+
''non-central  $  \chi  ^ {2} $-
 +
distribution''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669605.png" /> is the number of degrees of freedom and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669606.png" /> the parameter of non-centrality. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669607.png" /> this density is that of the ordinary (central) [[Chi-squared distribution| "chi-squared"  distribution]]. The [[Characteristic function|characteristic function]] of a non-central  "chi-squared"  distribution is
+
A continuous probability distribution concentrated on the positive semi-axis  $  0 < x < \infty $
 +
with density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669608.png" /></td> </tr></table>
+
$$
  
the [[Mathematical expectation|mathematical expectation]] and variance (cf. [[Dispersion|Dispersion]]) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n0669609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696010.png" />, respectively. A non-central  "chi-squared"  distribution belongs to the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]).
+
\frac{e ^ {- ( x + \lambda ) / 2 } x ^ {( n - 2 ) / 2 } }{2 ^ {n / 2 } \Gamma ( 1 / 2 ) }
  
As a rule, a non-central  "chi-squared"  distribution appears as the distribution of the sum of squares of independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696011.png" /> having normal distributions with non-zero means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696012.png" /> and unit variance; more precisely, the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696013.png" /> has a non-central  "chi-squared"  distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696014.png" /> degrees of freedom and non-centrality parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696015.png" />. The sum of several mutually independent random variables with a non-central "chi-squared" distribution has a distribution of the same type and its parameters are the sums of the corresponding parameters of the summands.
+
\sum _ {r = 0 } ^ \infty  
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696016.png" /> is even, then the distribution function of a non-central "chi-squared" distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696017.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696019.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696020.png" /> by
+
\frac{\lambda ^ {r} x ^ {r} }{( 2 r ) ! }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696021.png" /></td> </tr></table>
+
\frac{\Gamma ( r + 1 / 2 ) }{\Gamma ( r + n / 2 ) }
 +
,
 +
$$
  
This formula establishes a link between a non-central "chi-squared"  distribution and a [[Poisson distribution|Poisson distribution]]. Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696023.png" /> have Poisson distributions with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696025.png" />, respectively, then for any positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696026.png" />,
+
where  $  n $
 +
is the number of degrees of freedom and  $  \lambda $
 +
the parameter of non-centrality. For  $  \lambda = 0 $
 +
this density is that of the ordinary (central) [[Chi-squared distribution| "chi-squared"  distribution]]. The [[Characteristic function|characteristic function]] of a non-central  "chi-squared" distribution is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696027.png" /></td> </tr></table>
+
$$
 +
\phi ( t)  = ( 1 - 2 i t )  ^ {-} n/2  \mathop{\rm exp}
 +
\left \{
 +
\frac{\lambda i t }{1 - 2 i t }
 +
\right \} ;
 +
$$
 +
 
 +
the [[Mathematical expectation|mathematical expectation]] and variance (cf. [[Dispersion|Dispersion]]) are  $  n + \lambda $
 +
and  $  2 ( n + 2 \lambda ) $,
 +
respectively. A non-central  "chi-squared"  distribution belongs to the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]).
 +
 
 +
As a rule, a non-central  "chi-squared" distribution appears as the distribution of the sum of squares of independent random variables  $  X _ {1} \dots X _ {n} $
 +
having normal distributions with non-zero means  $  m _ {i} $
 +
and unit variance; more precisely, the sum  $  X _ {1}  ^ {2} + \dots X _ {n}  ^ {2} $
 +
has a non-central  "chi-squared"  distribution with  $  n $
 +
degrees of freedom and non-centrality parameter  $  \lambda = \sum _ {i=} 1  ^ {n} m _ {i}  ^ {2} $.
 +
The sum of several mutually independent random variables with a non-central  "chi-squared" distribution has a distribution of the same type and its parameters are the sums of the corresponding parameters of the summands.
 +
 
 +
If  $  n $
 +
is even, then the distribution function of a non-central  "chi-squared" distribution  $  F _ {n} ( x ; \lambda ) $
 +
is given by  $  F _ {n} ( x ; \lambda ) = 0 $
 +
for  $  x \leq  0 $
 +
and for  $  x > 0 $
 +
by
 +
 
 +
$$
 +
F _ {n} ( x ;  \lambda )  = \
 +
\sum _ { m= } 0 ^  \infty  \
 +
\sum _ {k = m + n / 2 } ^  \infty 
 +
 
 +
\frac{( \lambda / 2 )  ^ {m} ( x / 2 )  ^ {k} }{m ! k ! }
 +
 
 +
e ^ {- ( \lambda + x ) / 2 } .
 +
$$
 +
 
 +
This formula establishes a link between a non-central  "chi-squared"  distribution and a [[Poisson distribution|Poisson distribution]]. Namely, if  $  X $
 +
and  $  Y $
 +
have Poisson distributions with parameters  $  x / 2 $
 +
and  $  \lambda / 2 $,
 +
respectively, then for any positive integer  $  s > 0 $,
 +
 
 +
$$
 +
{\mathsf P} \{ X - Y \geq  s \}  =  F _ {2s} ( x ;  \lambda ) .
 +
$$
  
 
A non-central  "chi-squared"  distribution often arises in problems of mathematical statistics concerned with the study of the power of tests of  "chi-squared"  type. Since tables of non-central  "chi-squared"  distributions are fairly complete, various approximations by means of a  "chi-squared"  and a normal distribution are widely used in statistical applications.
 
A non-central  "chi-squared"  distribution often arises in problems of mathematical statistics concerned with the study of the power of tests of  "chi-squared"  type. Since tables of non-central  "chi-squared"  distributions are fairly complete, various approximations by means of a  "chi-squared"  and a normal distribution are widely used in statistical applications.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.B. Patnaik,  "The non-central <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696028.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696029.png" />-distributions and their applications"  ''Biometrica'' , '''36'''  (1949)  pp. 202–232</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,  A. Stuart,  "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.B. Patnaik,  "The non-central <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696028.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066960/n06696029.png" />-distributions and their applications"  ''Biometrica'' , '''36'''  (1949)  pp. 202–232</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics" , '''2. Continuous univariate distributions''' , Wiley  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics" , '''2. Continuous univariate distributions''' , Wiley  (1970)</TD></TR></table>

Revision as of 08:03, 6 June 2020


non-central $ \chi ^ {2} $- distribution

A continuous probability distribution concentrated on the positive semi-axis $ 0 < x < \infty $ with density

$$ \frac{e ^ {- ( x + \lambda ) / 2 } x ^ {( n - 2 ) / 2 } }{2 ^ {n / 2 } \Gamma ( 1 / 2 ) } \sum _ {r = 0 } ^ \infty \frac{\lambda ^ {r} x ^ {r} }{( 2 r ) ! } \frac{\Gamma ( r + 1 / 2 ) }{\Gamma ( r + n / 2 ) } , $$

where $ n $ is the number of degrees of freedom and $ \lambda $ the parameter of non-centrality. For $ \lambda = 0 $ this density is that of the ordinary (central) "chi-squared" distribution. The characteristic function of a non-central "chi-squared" distribution is

$$ \phi ( t) = ( 1 - 2 i t ) ^ {-} n/2 \mathop{\rm exp} \left \{ \frac{\lambda i t }{1 - 2 i t } \right \} ; $$

the mathematical expectation and variance (cf. Dispersion) are $ n + \lambda $ and $ 2 ( n + 2 \lambda ) $, respectively. A non-central "chi-squared" distribution belongs to the class of infinitely-divisible distributions (cf. Infinitely-divisible distribution).

As a rule, a non-central "chi-squared" distribution appears as the distribution of the sum of squares of independent random variables $ X _ {1} \dots X _ {n} $ having normal distributions with non-zero means $ m _ {i} $ and unit variance; more precisely, the sum $ X _ {1} ^ {2} + \dots X _ {n} ^ {2} $ has a non-central "chi-squared" distribution with $ n $ degrees of freedom and non-centrality parameter $ \lambda = \sum _ {i=} 1 ^ {n} m _ {i} ^ {2} $. The sum of several mutually independent random variables with a non-central "chi-squared" distribution has a distribution of the same type and its parameters are the sums of the corresponding parameters of the summands.

If $ n $ is even, then the distribution function of a non-central "chi-squared" distribution $ F _ {n} ( x ; \lambda ) $ is given by $ F _ {n} ( x ; \lambda ) = 0 $ for $ x \leq 0 $ and for $ x > 0 $ by

$$ F _ {n} ( x ; \lambda ) = \ \sum _ { m= } 0 ^ \infty \ \sum _ {k = m + n / 2 } ^ \infty \frac{( \lambda / 2 ) ^ {m} ( x / 2 ) ^ {k} }{m ! k ! } e ^ {- ( \lambda + x ) / 2 } . $$

This formula establishes a link between a non-central "chi-squared" distribution and a Poisson distribution. Namely, if $ X $ and $ Y $ have Poisson distributions with parameters $ x / 2 $ and $ \lambda / 2 $, respectively, then for any positive integer $ s > 0 $,

$$ {\mathsf P} \{ X - Y \geq s \} = F _ {2s} ( x ; \lambda ) . $$

A non-central "chi-squared" distribution often arises in problems of mathematical statistics concerned with the study of the power of tests of "chi-squared" type. Since tables of non-central "chi-squared" distributions are fairly complete, various approximations by means of a "chi-squared" and a normal distribution are widely used in statistical applications.

References

[1] 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)
[2] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1979)
[3] P.B. Patnaik, "The non-central - and -distributions and their applications" Biometrica , 36 (1949) pp. 202–232

Comments

References

[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 2. Continuous univariate distributions , Wiley (1970)
How to Cite This Entry:
Non-central chi-squared distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-central_chi-squared_distribution&oldid=28548
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article