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

:$Z = \sqrt$

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:

:$\lambda=\sqrt$

Properties

Probability density function

The probability density function (pdf) is

:$f(x;k,\lambda)=\frac I_(\lambda x)$

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

Raw moments

The first few raw moments are:

:$\mu^'_1=\sqrtL_^\left(\frac\right)$
:$\mu^'_2=k+\lambda^2$
:$\mu^'_3=3\sqrtL_^\left(\frac\right)$
:$\mu^'_4=(k+\lambda^2)^2+2(k+2\lambda^2)$

where $L_n^(z)$ 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 $X_j = (X_, X_), j = 1, 2, \dots n$, 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
:$E(X_j)= \mu=(\mu_1, \mu_2)^T, \qquad \Sigma = \begin \sigma_ & \sigma_ \\ \sigma_ & \sigma_ \end = \begin \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end,$
with $\Sigma$ positive definite. Define
:
U = \left \sum_^n \frac \right, \qquad
V = \left \sum_^n \frac \right.

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 σ''X'' follows a folded normal distribution whose parameters are equal to σλ and σ² for any value of σ.

References

{{DEFAULTSORT:Noncentral Chi Distribution
Continuous distributions
Noncentral distributions