Difference between revisions of "Non-central chi-squared 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 = \ | + | 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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$$ | $$ | ||
F _ {n} ( x ; \lambda ) = \ | F _ {n} ( x ; \lambda ) = \ | ||
− | \ | + | \sum_{m=0}^ \infty \ |
\sum _ {k = m + n / 2 } ^ \infty | \sum _ {k = m + n / 2 } ^ \infty | ||
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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><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> | |
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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) |
Non-central chi-squared distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-central_chi-squared_distribution&oldid=55311