Noncentral chi distribution

Noncentral chi
Parameters	$k>0\,$ degrees of freedom $\lambda >0\,$
Support	$x\in [0;+\infty )\,$
PDF	${\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)$
CDF	$1-Q_{\frac {k}{2}}\left(\lambda ,x\right)$ with Marcum Q-function $Q_{M}(a,b)$
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 _{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: $k$ 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 _{i=1}^{k}\left({\frac {\mu _{i}}{\sigma _{i}}}\right)^{2}}}

Properties

Probability density function

The probability density function (pdf) 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.

Raw moments

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 2 $n$ th moment is the same as the $n$ th moment of the noncentral chi-squared distribution with $\lambda$ being replaced by $\lambda ^{2}$ .

Differential equation

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

\left\{{\begin{array}{l}x^{2}f''(x)+\left(-kx+2x^{3}+x\right)f'(x)+f(x)\left(-x^{2}\left(\lambda ^{2}+k-2\right)+k+x^{4}-1\right)=0,\\f(1)=e^{-{\frac {\lambda ^{2}}{2}}-{\frac {1}{2}}}\lambda ^{1-{\frac {k}{2}}}I_{\frac {k-2}{2}}(\lambda ),f'(1)=e^{-{\frac {\lambda ^{2}}{2}}-{\frac {1}{2}}}\lambda ^{2-{\frac {k}{2}}}I_{\frac {k-4}{2}}(\lambda )\end{array}}\right\}

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

↑ Marakatha Krishnan (1967). "The Noncentral Bivariate Chi Distribution". SIAM Review. 9 (4): 708–714. doi:10.1137/1009111.
↑ 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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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