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 _{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 _{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 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}.

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.[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

  • 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

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

References

  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. JSTOR 2027477. 
 
[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]
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