Noncentral chi distribution

Noncentral chi

Parameters	$k>0\,$ degrees of freedom $\lambda >0\,$
Support	$x\in [0;+\infty )\,$
PDF	${\displaystyle {\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)}$
CDF	${\displaystyle 1-Q_{\frac {k}{2}}\left(\lambda ,x\right)}$ with Marcum Q-function $Q_{M}(a,b)$
Mean	${\displaystyle {\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)\,}$
Variance	${\displaystyle k+\lambda ^{2}-\mu ^{2}\,}$

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

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

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

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

Properties

Probability density function

The probability density function (pdf) is

${\displaystyle f(x;k,\lambda )={\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)}$

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

Raw moments

The first few raw moments are:

${\displaystyle \mu _{1}^{'}={\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)}$

${\displaystyle \mu _{2}^{'}=k+\lambda ^{2}}$

${\displaystyle \mu _{3}^{'}=3{\sqrt {\frac {\pi }{2}}}L_{3/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)}$

${\displaystyle \mu _{4}^{'}=(k+\lambda ^{2})^{2}+2(k+2\lambda ^{2})}$

where ${\displaystyle L_{n}^{(a)}(z)}$ is the generalized Laguerre polynomial. Note that the 2$n$th moment is the same as the $n$th moment of the noncentral chi-squared distribution with $\lambda$ being replaced by $\lambda ^{2}$.

Differential equation

The pdf of the noncentral chi distribution is a solution to the following differential equation:

${\displaystyle \left\{{\begin{array}{l}x^{2}f''(x)+\left(-kx+2x^{3}+x\right)f'(x)+f(x)\left(-x^{2}\left(\lambda ^{2}+k-2\right)+k+x^{4}-1\right)=0,\\f(1)=e^{-{\frac {\lambda ^{2}}{2}}-{\frac {1}{2}}}\lambda ^{1-{\frac {k}{2}}}I_{\frac {k-2}{2}}(\lambda ),f'(1)=e^{-{\frac {\lambda ^{2}}{2}}-{\frac {1}{2}}}\lambda ^{2-{\frac {k}{2}}}I_{\frac {k-4}{2}}(\lambda )\end{array}}\right\}}$

Bivariate non-central chi distribution

Let ${\displaystyle X_{j}=(X_{1j},X_{2j}),j=1,2,\dots n}$, be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions ${\displaystyle N(\mu _{i},\sigma _{i}^{2}),i=1,2}$, correlation $\rho$, and mean vector and covariance matrix

${\displaystyle E(X_{j})=\mu =(\mu _{1},\mu _{2})^{T},\qquad \Sigma ={\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{21}&\sigma _{22}\end{bmatrix}}={\begin{bmatrix}\sigma _{1}^{2}&\rho \sigma _{1}\sigma _{2}\\\rho \sigma _{1}\sigma _{2}&\sigma _{2}^{2}\end{bmatrix}},}$

with $\Sigma$ positive definite. Define

${\displaystyle U=\left[\sum _{j=1}^{n}{\frac {X_{1j}^{2}}{\sigma _{1}^{2}}}\right]^{1/2},\qquad V=\left[\sum _{j=1}^{n}{\frac {X_{2j}^{2}}{\sigma _{2}^{2}}}\right]^{1/2}.}$

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom.^[1][2] If either or both ${\displaystyle \mu _{1}\neq 0}$ or ${\displaystyle \mu _{2}\neq 0}$ the distribution is a noncentral bivariate chi distribution.

Related distributions

• If $X$ is a random variable with the non-central chi distribution, the random variable $X^{2}$ will have the noncentral chi-squared distribution. Other related distributions may be seen there.
• If $X$ is chi distributed: ${\displaystyle X\sim \chi _{k}}$ then $X$ is also non-central chi distributed: ${\displaystyle X\sim NC\chi _{k}(0)}$. In other words, the chi distribution is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).
• A noncentral chi distribution with 2 degrees of freedom is equivalent to a Rice distribution with ${\displaystyle \sigma =1}$.
• If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution whose parameters are equal to σλ and σ² for any value of σ.

Applications

The Euclidean norm of a multivariate normally distributed random vector follows a noncentral chi distribution.

References

1. Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. doi:10.1137/1009111.
2. P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg (1963). "A note on the bivariate chi distribution". SIAM Review. 5: 140–144. doi:10.1137/1005034. JSTOR 2027477.