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Difference between revisions of "Non-central chi-squared distribution"

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m (AUTOMATIC EDIT (latexlist): Replaced 2 formulas out of 2 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
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distribution''
 
distribution''
  
A continuous probability distribution concentrated on the positive semi-axis  $  0 < x < \infty $
+
A continuous probability distribution concentrated on the positive semi-axis  $  0 < x < \infty $
 
with density
 
with density
  
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$$  
 
$$  
\phi ( t)  =  ( 1 - 2 i t )  ^ {-} n/2  \mathop{\rm exp}
+
\phi ( t)  =  ( 1 - 2 i t )  ^ {-n/2} \mathop{\rm exp}
 
\left \{  
 
\left \{  
 
\frac{\lambda i t }{1 - 2 i t }
 
\frac{\lambda i t }{1 - 2 i t }
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$$
 
$$
  
the [[Mathematical expectation|mathematical expectation]] and variance (cf. [[Dispersion|Dispersion]]) are  $  n + \lambda $
+
the [[mathematical expectation]] and variance (cf. [[Dispersion]]) are  $  n + \lambda $
 
and  $  2 ( n + 2 \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]]).
 
respectively. A non-central  "chi-squared"  distribution belongs to the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]).
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and unit variance; more precisely, the sum  $  X _ {1}  ^ {2} + \dots X _ {n}  ^ {2} $
 
and unit variance; more precisely, the sum  $  X _ {1}  ^ {2} + \dots X _ {n}  ^ {2} $
 
has a non-central  "chi-squared"  distribution with  $  n $
 
has a non-central  "chi-squared"  distribution with  $  n $
degrees of freedom and non-centrality parameter  $  \lambda = \sum _ {i=} ^ {n} m _ {i}  ^ {2} $.  
+
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.
 
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.
  
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is given by  $  F _ {n} ( x ;  \lambda ) = 0 $
 
is given by  $  F _ {n} ( x ;  \lambda ) = 0 $
 
for  $  x \leq  0 $
 
for  $  x \leq  0 $
and for  $  x &gt; 0 $
+
and for  $  x > 0 $
 
by
 
by
  
 
$$  
 
$$  
 
F _ {n} ( x ;  \lambda )  = \  
 
F _ {n} ( x ;  \lambda )  = \  
\sum _ { m= } 0 ^  \infty  \  
+
\sum_{m=0}^  \infty  \  
 
\sum _ {k = m + n / 2 } ^  \infty   
 
\sum _ {k = m + n / 2 } ^  \infty   
  
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have Poisson distributions with parameters  $  x / 2 $
 
have Poisson distributions with parameters  $  x / 2 $
 
and  $  \lambda / 2 $,  
 
and  $  \lambda / 2 $,  
respectively, then for any positive integer  $  s &gt; 0 $,
+
respectively, then for any positive integer  $  s > 0 $,
  
 
$$  
 
$$  
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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 $\chi ^ { 2 }$- and $F$-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 $\chi ^ { 2 }$- and $F$-distributions and their applications"  ''Biometrica'' , '''36'''  (1949)  pp. 202–232</td></tr>
 
+
<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>
====Comments====
 
 
 
====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>
 

Latest revision as of 18:53, 24 January 2024


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 $\chi ^ { 2 }$- and $F$-distributions and their applications" Biometrica , 36 (1949) pp. 202–232
[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=50478
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article