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Noncentral chi distribution

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

Definition

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:

Properties

Probability density function

The probability density function (pdf) is

where is a modified Bessel function of the first kind.

Raw moments

The first few raw moments are:

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

Bivariate non-central chi distribution

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. If either or both or the distribution is a noncentral bivariate chi distribution.

Related distributions

  • 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 σ<sup>2</sup> for any value of σ.

References