Noncentral chi distribution

In probability theory and statistics, the noncentral chi distribution[1] is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.

Noncentral chi
Parameters

degrees of freedom

Support
PDF
CDF with Marcum Q-function
Mean
Variance , where is the mean

DefinitionEdit

If   are k independent, normally distributed random variables with means   and variances  , then the statistic

 

is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters:   which specifies the number of degrees of freedom (i.e. the number of  ), and   which is related to the mean of the random variables   by:

 

PropertiesEdit

Probability density functionEdit

The probability density function (pdf) is

 

where   is a modified Bessel function of the first kind.

Raw momentsEdit

The first few raw moments are:

 
 
 
 

where   is a Laguerre function. Note that the 2 th moment is the same as the  th moment of the noncentral chi-squared distribution with   being replaced by  .

Bivariate non-central chi distributionEdit

Let  , be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions  , correlation  , and mean vector and covariance matrix

 

with   positive definite. Define

 

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom.[2][3] If either or both   or   the distribution is a noncentral bivariate chi distribution.

Related distributionsEdit

  • If   is a random variable with the non-central chi distribution, the random variable   will have the noncentral chi-squared distribution. Other related distributions may be seen there.
  • If   is chi distributed:   then   is also non-central chi distributed:  . 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  .
  • 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.

ReferencesEdit

  1. ^ J. H. Park (1961). "Moments of the Generalized Rayleigh Distribution". Quarterly of Applied Mathematics. 19 (1): 45–49. doi:10.1090/qam/119222.
  2. ^ Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. doi:10.1137/1009111.
  3. ^ 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.CS1 maint: multiple names: authors list (link)