Noncentral chi distribution

Noncentral chi

Parameters	$k > 0\,$ degrees of freedom $\lambda > 0\,$
Support	$x \in [0; +\infty)\,$
PDF	$\frac{e^{-(x^2+\lambda^2)/2}x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)$
CDF	$1 - Q_{\frac{k}{2}} \left( \lambda, x \right)$ 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	$k+\lambda^2-\mu^2\,$

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

$Z = \sqrt{\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2}$

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:

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

$\mu^'_2=k+\lambda^2$

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

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

where $L_n^{(a)}(z)$ 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:

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

Bivariate non-central chi distribution

Let $X_j = (X_{1j}, X_{2j}), j = 1, 2, \dots n$, 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

$E(X_j)= \mu=(\mu_1, \mu_2)^T, \qquad \Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{bmatrix} = \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix},$

with $\Sigma$ positive definite. Define

$U = \left[ \sum_{j=1}^n \frac{X_{1j}^2}{\sigma_1^2} \right]^{1/2}, \qquad V = \left[ \sum_{j=1}^n \frac{X_{2j}^2}{\sigma_2^2} \right]^{1/2}.$

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 $\mu_1 \neq 0$ or $\mu_2 \neq 0$ 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: $X \sim \chi_k$ 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.

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.