Noncentral chi distribution

Noncentral chi

Parameters	$k>0$ degrees of freedom $\lambda >0$
Support	$x\in [0;+\mathrm{\infty })$
PDF	$\frac{e^{-(x^2+{\lambda }^2)/2}{x}^k\lambda }{(\lambda x{)}^{k/2}}{I}_{k/2-1}(\lambda x)$
CDF	$1-Q_{\frac{k}{2}}(\lambda ,x)$ with Marcum Q-function $Q_M(a,b)$
Mean	$\sqrt{\frac{\pi }{2}}{L}_{1/2}^{(k/2-1)}\left(\frac{-{\lambda }^2}{2}\right)$
Variance	$k+{\lambda }^2-{\mu }^2$, where $\mu$ is the mean

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

Definition

If $X_i$ are k independent, normally distributed random variables with means ${\mu }_i$ and variances ${\sigma }_i^2$, then the statistic

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

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:

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

Properties

Probability density function

The probability density function (pdf) is

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

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

Raw moments

The first few raw moments are:

${\mu }_1^{\text{'}}=\sqrt{\frac{\pi }{2}}{L}_{1/2}^{(k/2-1)}\left(\frac{-{\lambda }^2}{2}\right)$

${\mu }_2^{\text{'}}=k+{\lambda }^2$

${\mu }_3^{\text{'}}=3\sqrt{\frac{\pi }{2}}{L}_{3/2}^{(k/2-1)}\left(\frac{-{\lambda }^2}{2}\right)$

${\mu }_4^{\text{'}}=(k+{\lambda }^2{)}^2+2(k+2{\lambda }^2)$

where $L_n^{(a)}(z)$ is a Laguerre function. 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 ${\lambda }^2$.

Bivariate non-central chi distribution

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

$E(X_j)=\mu =({\mu }_1,{\mu }_2{)}^T,\mathrm{\Sigma }=\left[\begin{array}{cc}{\sigma }_{11}& {\sigma }_{12}\\ {\sigma }_{21}& {\sigma }_{22}\end{array}\right]=\left[\begin{array}{cc}{\sigma }_1^2& \rho {\sigma }_1{\sigma }_2\\ \rho {\sigma }_1{\sigma }_2& {\sigma }_2^2\end{array}\right],$

with $\mathrm{\Sigma }$ positive definite. Define

$U={\left[\sum _{j=1}^n\frac{X_{1j}^2}{{\sigma }_1^2}\right]}^{1/2},V={\left[\sum _{j=1}^n\frac{X_{2j}^2}{{\sigma }_2^2}\right]}^{1/2}.$

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom. If either or both ${\mu }_1\ne 0$ or ${\mu }_2\ne 0$ 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: $X\sim {\chi }_k$ then $X$ is also non-central chi distributed: $X\sim NC{\chi }_k(0)$. 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 $\sigma =1$.
• 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.