Noncentral chi distribution

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)
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 Xi are k independent, normally distributed random variables with means μi and variances \sigma_i^2, then the statistic

Z = \sqrt{\sum_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 Xi), and λ which is related to the mean of the random variables Xi by:

\lambda=\sqrt{\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2}

Properties

The probability density function 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 Iν(z) is a modified Bessel function of the first kind.

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 2nth moment is the same as the nth moment of the noncentral chi-squared distribution with λ being replaced by λ2.

Related distributions

  • If X is a random variable with the non-central chi distribution, the random variable X2 will have the noncentral chi-squared distribution. Other related distributions may be seen there.
  • If X is chi distributed: X∼χk then X is also non-central chi distributed: XNCχk(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 σ = 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.

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