Non-central chi-squared distribution

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)