Noncentral chi 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=\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 2nth moment is the same as the nth 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.

References

References

1. J. H. Park. (1961). "Moments of the Generalized Rayleigh Distribution". Quarterly of Applied Mathematics.
2. Marakatha Krishnan. (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review.
3. P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg. (1963). "A note on the bivariate chi distribution". SIAM Review.