Noncentral chi distribution

In probability theory and statistics, the noncentral chi distribution^[1] is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.

Noncentral chi

Parameters	${\displaystyle k>0\,}$ degrees of freedom ${\displaystyle \lambda >0\,}$
Support	${\displaystyle x\in [0;+\infty )\,}$
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 ${\displaystyle Q_{M}(a,b)}$
Mean	${\displaystyle {\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)\,}$
Variance	${\displaystyle k+\lambda ^{2}-\mu ^{2}}$, where ${\displaystyle \mu }$ is the mean

Definition

If ${\displaystyle X_{i}}$  are k independent, normally distributed random variables with means ${\displaystyle \mu _{i}}$  and variances ${\displaystyle \sigma _{i}^{2}}$ , 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: ${\displaystyle k}$  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:

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

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 a Laguerre function. Note that the 2${\displaystyle n}$ th moment is the same as the ${\displaystyle n}$ th moment of the noncentral chi-squared distribution with ${\displaystyle \lambda }$  being replaced by ${\displaystyle \lambda ^{2}}$ .

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 ${\displaystyle \rho }$ , 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 ${\displaystyle \Sigma }$  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.^[2][3] 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 ${\displaystyle X}$  is a random variable with the non-central chi distribution, the random variable ${\displaystyle X^{2}}$  will have the noncentral chi-squared distribution. Other related distributions may be seen there.
• If ${\displaystyle X}$  is chi distributed: ${\displaystyle X\sim \chi _{k}}$  then ${\displaystyle X}$  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.

References

1. J. H. Park (1961). "Moments of the Generalized Rayleigh Distribution". Quarterly of Applied Mathematics. 19 (1): 45–49. doi:10.1090/qam/119222. JSTOR 43634840.
2. Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. Bibcode:1967SIAMR...9..708K. doi:10.1137/1009111.
3. P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg (1963). "A note on the bivariate chi distribution". SIAM Review. 5 (2): 140–144. Bibcode:1963SIAMR...5..140K. doi:10.1137/1005034. JSTOR 2027477.{{cite journal}}: CS1 maint: multiple names: authors list (link)