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Definition

If X i {\displaystyle X_{i}} are k independent, normally distributed random variables with means μ i {\displaystyle \mu _{i}} and variances σ i 2 {\displaystyle \sigma _{i}^{2}} , then the statistic

Z = ∑ i = 1 k ( X i σ i ) 2 {\displaystyle 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 {\displaystyle k} which specifies the number of degrees of freedom (i.e. the number of X i {\displaystyle X_{i}} ), and λ {\displaystyle \lambda } which is related to the mean of the random variables X i {\displaystyle X_{i}} by:

λ = ∑ i = 1 k ( μ i σ i ) 2 {\displaystyle \lambda ={\sqrt {\sum _{i=1}^{k}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}}}}

Properties

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 ) {\displaystyle 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 ν ( z ) {\displaystyle I_{\nu }(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 ) {\displaystyle \mu _{1}^{'}={\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)} μ 2 ′ = k + λ 2 {\displaystyle \mu _{2}^{'}=k+\lambda ^{2}} μ 3 ′ = 3 π 2 L 3 / 2 ( k / 2 − 1 ) ( − λ 2 2 ) {\displaystyle \mu _{3}^{'}=3{\sqrt {\frac {\pi }{2}}}L_{3/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)} μ 4 ′ = ( k + λ 2 ) 2 + 2 ( k + 2 λ 2 ) {\displaystyle \mu _{4}^{'}=(k+\lambda ^{2})^{2}+2(k+2\lambda ^{2})}

where L n ( a ) ( z ) {\displaystyle L_{n}^{(a)}(z)} is a Laguerre function. Note that the 2 n {\displaystyle n} th moment is the same as the n {\displaystyle n} th moment of the noncentral chi-squared distribution with λ {\displaystyle \lambda } being replaced by λ 2 {\displaystyle \lambda ^{2}} .

Bivariate non-central chi distribution

Let X j = ( X 1 j , X 2 j ) , j = 1 , 2 , … n {\displaystyle 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 ( μ i , σ i 2 ) , i = 1 , 2 {\displaystyle N(\mu _{i},\sigma _{i}^{2}),i=1,2} , correlation ρ {\displaystyle \rho } , 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 ] , {\displaystyle 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 Σ {\displaystyle \Sigma } 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 . {\displaystyle 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.²³ If either or both μ 1 ≠ 0 {\displaystyle \mu _{1}\neq 0} or μ 2 ≠ 0 {\displaystyle \mu _{2}\neq 0} the distribution is a noncentral bivariate chi distribution.

Related distributions

• If X {\displaystyle X} is a random variable with the non-central chi distribution, the random variable X 2 {\displaystyle X^{2}} will have the noncentral chi-squared distribution. Other related distributions may be seen there.
• If X {\displaystyle X} is chi distributed: X ∼ χ k {\displaystyle X\sim \chi _{k}} then X {\displaystyle X} is also non-central chi distributed: X ∼ N C χ k ( 0 ) {\displaystyle 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 σ = 1 {\displaystyle \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.

References

1. J. H. Park (1961). "Moments of the Generalized Rayleigh Distribution". Quarterly of Applied Mathematics. 19 (1): 45–49. doi:10.1090/qam/119222. JSTOR 43634840. https://doi.org/10.1090%2Fqam%2F119222 ↩

2. Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. Bibcode:1967SIAMR...9..708K. doi:10.1137/1009111. /wiki/Bibcode_(identifier) ↩

3. P. R. Krishnaiah, P. Hagis, Jr. and L. Steinberg (1963). "A note on the bivariate chi distribution". SIAM Review. 5 (2): 140–144. Bibcode:1963SIAMR...5..140K. doi:10.1137/1005034. JSTOR 2027477.{{cite journal}}: CS1 maint: multiple names: authors list (link) /wiki/Bibcode_(identifier) ↩