Noncentral chi distribution

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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 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[edit]

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 I_\nu(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 \lambda being replaced by \lambda^2.

Differential equation



\left\{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 )\right\}

Bivariate non-central chi distribution[edit]

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[edit]

  • 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[edit]

The Euclidean norm of a multivariate normally distributed random vector follows a noncentral chi distribution.

References[edit]

  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.