# Noncentral chi-square distribution

Parameters Probability density function325px Cumulative distribution function325px ${\displaystyle k>0\,}$ degrees of freedom ${\displaystyle \lambda >0\,}$ non-centrality parameter ${\displaystyle x\in [0;+\infty )\,}$ ${\displaystyle {\frac {1}{2}}e^{-(x+\lambda )/2}\left({\frac {x}{\lambda }}\right)^{k/4-1/2}I_{k/2-1}({\sqrt {\lambda x}})}$ :${\displaystyle \sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {(\lambda /2)^{j}}{j!}}{\frac {\gamma (j+k/2,x/2)}{\Gamma (j+k/2)}}\,}$ ${\displaystyle k+\lambda \,}$ ${\displaystyle 2(k+2\lambda )\,}$ ${\displaystyle {\frac {2^{3/2}(k+3\lambda )}{(k+2\lambda )^{3/2}}}}$ ${\displaystyle {\frac {12(k+4\lambda )}{(k+2\lambda )^{2}}}}$ ${\displaystyle {\frac {\exp \left({\frac {\lambda t}{1-2t}}\right)}{(1-2t)^{k/2}}}}$ ${\displaystyle {\frac {\exp \left({\frac {i\lambda t}{1-2it}}\right)}{(1-2it)^{k/2}}}}$

In probability theory and statistics, the noncentral chi-square or noncentral ${\displaystyle \chi ^{2}}$ distribution is a generalization of the chi-square distribution. If ${\displaystyle X_{i}}$ are k independent, normally distributed random variables with means ${\displaystyle \mu _{i}}$ and variances ${\displaystyle \sigma _{i}^{2}}$, then the random variable

${\displaystyle \sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}$

is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution has two parameters: ${\displaystyle k}$ which specifies the number of degrees of freedom (i.e. the number of ${\displaystyle X_{i}}$), and ${\displaystyle \lambda }$ which is related to the mean of the random variables ${\displaystyle X_{i}}$ by:

${\displaystyle \lambda =\sum _{1}^{k}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}.}$

Note that some references define ${\displaystyle \lambda }$ as one half of the above sum.

## Properties

The noncentral chi-square distribution is equivalent to a (central) chi-square distribution with ${\displaystyle k+2P}$ degrees of freedom, where ${\displaystyle P}$ is a Poisson random variable with parameter ${\displaystyle \lambda /2}$. Thus, the probability density function is given by

${\displaystyle f_{X}(x;k,\lambda )=\sum _{i=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{i}}{i!}}f_{Y_{k+2i}}(x),}$

where ${\displaystyle Y_{q}}$ is distributed as chi-square with ${\displaystyle q}$ degrees of freedom.

Alternatively, the pdf can be written as

${\displaystyle f_{X}(x;k,\lambda )={\frac {1}{2}}e^{-(x+\lambda )/2}\left({\frac {x}{\lambda }}\right)^{k/4-1/2}I_{k/2-1}({\sqrt {\lambda x}})}$

where ${\displaystyle I_{\nu }(z)}$ is a modified Bessel function of the first kind given by

${\displaystyle I_{a}(y):=(y/2)^{a}\sum _{j=0}^{\infty }{\frac {(y^{2}/4)^{j}}{j!\Gamma (a+j+1)}}}$

The moment generating function is given by

${\displaystyle M(t;k,\lambda )={\frac {\exp \left({\frac {\lambda t}{1-2t}}\right)}{(1-2t)^{k/2}}}.}$

The first few raw moments are:

${\displaystyle \mu _{1}^{'}=k+\lambda }$
${\displaystyle \mu _{2}^{'}=(k+\lambda )^{2}+2(k+2\lambda )}$
${\displaystyle \mu _{3}^{'}=(k+\lambda )^{3}+6(k+\lambda )(k+2\lambda )+8(k+3\lambda )}$
${\displaystyle \mu _{4}^{'}=(k+\lambda )^{4}+12(k+\lambda )^{2}(k+2\lambda )+4(11k^{2}+44k\lambda +36\lambda ^{2})+48(k+4\lambda )}$

The first few central moments are:

${\displaystyle \mu _{2}=2(k+2\lambda )\,}$
${\displaystyle \mu _{3}=8(k+3\lambda )\,}$
${\displaystyle \mu _{4}=12(k+2\lambda )^{2}+48(k+4\lambda )\,}$

The nth cumulant is

${\displaystyle K_{n}=2^{n-1}(n-1)!(k+n\lambda ).\,}$

Hence

${\displaystyle \mu _{n}^{'}=2^{n-1}(n-1)!(k+n\lambda )+\sum _{j=1}^{n-1}{\frac {(n-1)!2^{j-1}}{(n-j)!}}(k+j\lambda )\mu _{n-j}^{'}}$

Again using the relation between the central and noncentral chi-square distributions, the cumulative distribution function (cdf) can be written as

${\displaystyle P(x;k,\lambda )=\sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {(\lambda /2)^{j}}{j!}}Q(x;k+2j)}$

where ${\displaystyle Q(x;k)}$ is the cumulative density of the central chi-squared distribution which is given by

${\displaystyle Q(x;k)={\frac {\gamma (k/2,x/2)}{\Gamma (k/2)}}\,}$

where ${\displaystyle \gamma (k,z)}$ is the lower incomplete Gamma function.

## Derivation of the pdf

The derivation of the probability density function is most easily done by performing the following steps:

1. Start with the joint PDF of two independent non-zero mean Gaussian distributions, ${\displaystyle X}$ and ${\displaystyle Y}$.
2. Convert the joint density ${\displaystyle f(X,Y)}$ to polar: ${\displaystyle f(R,A)}$ where ${\displaystyle R^{2}=(X^{2}+Y^{2})}$, ${\displaystyle tan(A)=Y/X}$.
3. Integrate over the angular variable ${\displaystyle A}$.
4. Convert from R to r where ${\displaystyle r^{2}=R}$. This will yield a series expansion in r one factor of which matches the modified Bessel function ${\displaystyle I_{0}}$.
5. Take the Laplace (Fourier) transform term-by-term and the special case K = 2 and the MGF will result.
6. For the general case, take the K = 2 MGF and raise it to the ${\displaystyle K/2}$ power.
7. The final trick to hide the K-dependence in the numerator of the MGF is to note that ${\displaystyle \lambda }$ is a function of K; that is,
${\displaystyle \lambda _{2}=\sum _{1}^{2}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}}$
${\displaystyle \lambda _{K}=\sum _{1}^{k}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}=\lambda }$
and therefore, ${\displaystyle \lambda }$ is not explicitly a function of K in the above table.

## Related distributions

• If ${\displaystyle Z}$ is chi-square distributed ${\displaystyle Z\sim \chi _{k}^{2}}$ then ${\displaystyle Z}$ is also non-central chi-square distributed: ${\displaystyle Z\sim {\chi '}_{k}^{2}(0)}$
• If ${\displaystyle J\sim Poisson(\lambda /2)}$, then ${\displaystyle {\chi '}_{k}^{2}(\lambda )\sim \chi _{k+2J}^{2}}$
Various chi and chi-square distributions
Name Statistic
chi-square distribution ${\displaystyle \sum _{1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}$
noncentral chi-square distribution ${\displaystyle \sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}$
chi distribution ${\displaystyle {\sqrt {\sum _{1}^{k}\left({\frac {X_{i}-\mu _{i}}{\sigma _{i}}}\right)^{2}}}}$
noncentral chi distribution ${\displaystyle {\sqrt {\sum _{1}^{k}\left({\frac {X_{i}}{\sigma _{i}}}\right)^{2}}}}$

## References

• Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
• Johnson, N. L. and Kotz, S., (1970), Continuous Univariate Distributions, vol. 2, Houghton-Mifflin.