Noncentral chi distribution

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:


 * $$\lambda=\sqrt{\sum_{i=1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2}$$

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 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, \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}. $$ Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom. If either or both $$\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.