Noncentral chi distribution

Noncentral chi

Parameters	[math]\displaystyle{ k \gt 0\, }[/math] degrees of freedom [math]\displaystyle{ \lambda \gt 0\, }[/math]
Support	[math]\displaystyle{ x \in [0; +\infty)\, }[/math]
PDF	[math]\displaystyle{ \frac{e^{-(x^2+\lambda^2)/2}x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x) }[/math]
CDF	[math]\displaystyle{ 1 - Q_{\frac{k}{2}} \left( \lambda, x \right) }[/math] with Marcum Q-function [math]\displaystyle{ Q_M(a,b) }[/math]
Mean	[math]\displaystyle{ \sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right)\, }[/math]
Variance	[math]\displaystyle{ k+\lambda^2-\mu^2 }[/math], where [math]\displaystyle{ \mu }[/math] 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 [math]\displaystyle{ X_i }[/math] are k independent, normally distributed random variables with means [math]\displaystyle{ \mu_i }[/math] and variances [math]\displaystyle{ \sigma_i^2 }[/math], then the statistic

[math]\displaystyle{ Z = \sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2} }[/math]

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

[math]\displaystyle{ \lambda=\sqrt{\sum_{i=1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2} }[/math]

Properties

Probability density function

The probability density function (pdf) is

[math]\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) }[/math]

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

Raw moments

The first few raw moments are:

[math]\displaystyle{ \mu^'_1=\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right) }[/math]

[math]\displaystyle{ \mu^'_2=k+\lambda^2 }[/math]

[math]\displaystyle{ \mu^'_3=3\sqrt{\frac{\pi}{2}}L_{3/2}^{(k/2-1)}\left(\frac{-\lambda^2}{2}\right) }[/math]

[math]\displaystyle{ \mu^'_4=(k+\lambda^2)^2+2(k+2\lambda^2) }[/math]

where [math]\displaystyle{ L_n^{(a)}(z) }[/math] is a Laguerre function. Note that the 2[math]\displaystyle{ n }[/math]th moment is the same as the [math]\displaystyle{ n }[/math]th moment of the noncentral chi-squared distribution with [math]\displaystyle{ \lambda }[/math] being replaced by [math]\displaystyle{ \lambda^2 }[/math].

Bivariate non-central chi distribution

Let [math]\displaystyle{ X_j = (X_{1j}, X_{2j}), j = 1, 2, \dots n }[/math], be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions [math]\displaystyle{ N(\mu_i,\sigma_i^2), i=1,2 }[/math], correlation [math]\displaystyle{ \rho }[/math], and mean vector and covariance matrix

[math]\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}, }[/math]

with [math]\displaystyle{ \Sigma }[/math] positive definite. Define

[math]\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}. }[/math]

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom. If either or both [math]\displaystyle{ \mu_1 \neq 0 }[/math] or [math]\displaystyle{ \mu_2 \neq 0 }[/math] the distribution is a noncentral bivariate chi distribution.

Related distributions

• If [math]\displaystyle{ X }[/math] is a random variable with the non-central chi distribution, the random variable [math]\displaystyle{ X^2 }[/math] will have the noncentral chi-squared distribution. Other related distributions may be seen there.
• If [math]\displaystyle{ X }[/math] is chi distributed: [math]\displaystyle{ X \sim \chi_k }[/math] then [math]\displaystyle{ X }[/math] is also non-central chi distributed: [math]\displaystyle{ X \sim NC\chi_k(0) }[/math]. 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 [math]\displaystyle{ \sigma=1 }[/math].
• 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.