Noncentral chi distribution - Alchetron, the free social encyclopedia


Parameters k > 0 {\displaystyle k>0\,} degrees of freedom λ > 0 {\displaystyle \lambda >0\,} Support x ∈ [ 0 ; + ∞ ) {\displaystyle x\in [0;+\infty )\,} PDF e − ( x 2 + λ 2 ) / 2 x k λ ( λ x ) k / 2 I k / 2 − 1 ( λ x ) {\displaystyle {\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)} CDF 1 − Q k 2 ( λ , x ) {\displaystyle 1-Q_{\frac {k}{2}}\left(\lambda ,x\right)} with Marcum Q-function Q M ( a , b ) {\displaystyle Q_{M}(a,b)} Mean π 2 L 1 / 2 ( k / 2 − 1 ) ( − λ 2 2 ) {\displaystyle {\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)\,} Variance k + λ 2 − μ 2 {\displaystyle 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 μ i and variances σ i 2 , then the statistic

Probability density function

The probability density function (pdf) is

f ( x ; k , λ ) = e − ( x 2 + λ 2 ) / 2 x k λ ( λ x ) k / 2 I k / 2 − 1 ( λ x )

where I ν ( z ) is a modified Bessel function of the first kind.

Raw moments

The first few raw moments are:

μ 1 ′ = π 2 L 1 / 2 ( k / 2 − 1 ) ( − λ 2 2 ) μ 2 ′ = k + λ 2 μ 3 ′ = 3 π 2 L 3 / 2 ( k / 2 − 1 ) ( − λ 2 2 ) μ 4 ′ = ( k + λ 2 ) 2 + 2 ( k + 2 λ 2 )

where L n ( a ) ( z ) is the generalized Laguerre polynomial. Note that the 2 n th moment is the same as the n th moment of the noncentral chi-squared distribution with λ being replaced by λ 2 .

Differential equation

The pdf of the noncentral chi distribution is a solution to the following differential equation:

{ x 2 f ″ ( x ) + ( − k x + 2 x 3 + x ) f ′ ( x ) + f ( x ) ( − x 2 ( λ 2 + k − 2 ) + k + x 4 − 1 ) = 0 , f ( 1 ) = e − λ 2 2 − 1 2 λ 1 − k 2 I k − 2 2 ( λ ) , f ′ ( 1 ) = e − λ 2 2 − 1 2 λ 2 − k 2 I k − 4 2 ( λ ) }

Bivariate non-central chi distribution

Let X j = ( X 1 j , X 2 j ) , j = 1 , 2 , … n , be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions N ( μ i , σ i 2 ) , i = 1 , 2 , correlation ρ , and mean vector and covariance matrix

E ( X j ) = μ = ( μ 1 , μ 2 ) T , Σ = [ σ 11 σ 12 σ 21 σ 22 ] = [ σ 1 2 ρ σ 1 σ 2 ρ σ 1 σ 2 σ 2 2 ] ,

with Σ positive definite. Define

U = [ ∑ j = 1 n X 1 j 2 σ 1 2 ] 1 / 2 , V = [ ∑ j = 1 n X 2 j 2 σ 2 2 ] 1 / 2 .

Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom. If either or both μ 1 ≠ 0 or μ 2 ≠ 0 the distribution is a noncentral bivariate chi distribution.

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 ∼ χ k then X is also non-central chi distributed: X ∼ N C χ 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 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.

Applications

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

References

Noncentral chi distribution Wikipedia

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