Noncentral chi-squared distribution

Noncentral chi-squared
Probability density function
Cumulative distribution function
Parameters	$k > 0\,$ degrees of freedom $\lambda > 0\,$ non-centrality parameter
Support	$x \in [0; +\infty)\,$
PDF	$\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$
CDF	$1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)$ with Marcum Q-function $Q_M(a,b)$
Mean	$k+\lambda\,$
Variance	$2(k+2\lambda)\,$
Skewness	$\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}$
Ex. kurtosis	$\frac{12(k+4\lambda)}{(k+2\lambda)^2}$
MGF	$\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}} \text{ for }2t<1$
CF	$\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}$

In probability theory and statistics, the noncentral chi-squared or noncentral $\chi^2$ distribution is a generalization of the chi-squared distribution. This distribution often arises in the power analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chi-squared distribution; important examples of such tests are the likelihood ratio tests.

Background

Let ( $X_1$ , $X_2, \ldots,$ $X_i, \ldots,$ $X_k$ ) be k independent, normally distributed random variables with means $\mu_i$ and unit variances. Then the random variable

\sum_{i=1}^k X_i^2

is distributed according to the noncentral chi-squared distribution. It 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=\sum_{i=1}^k \mu_i^2.

$\lambda$ is sometimes called the noncentrality parameter. Note that some references define $\lambda$ in other ways, such as half of the above sum, or its square root.

This distribution arises in multivariate statistics as a derivative of the multivariate normal distribution. While the central chi-squared distribution is the squared norm of a random vector with $N(0_k,I_k)$ distribution (i.e., the squared distance from the origin of a point taken at random from that distribution), the non-central $\chi^2$ is the squared norm of a random vector with $N(\mu,I_k)$ distribution. Here $0_k$ is a zero vector of length k, $\mu = (\mu_1, \ldots, \mu_k)$ and $I_k$ is the identity matrix of size k.

Definition

The probability density function (pdf) is given by

f_X(x; k,\lambda) = \sum_{i=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^i}{i!} f_{Y_{k+2i}}(x),

where $Y_q$ is distributed as chi-squared with $q$ degrees of freedom.

From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions. Suppose that a random variable J has a Poisson distribution with mean $\lambda/2$ , and the conditional distribution of Z given $J=i$ is chi-squared with k+2i degrees of freedom. Then the unconditional distribution of Z is non-central chi-squared with k degrees of freedom, and non-centrality parameter $\lambda$ .

Alternatively, the pdf can be written as

f_X(x;k,\lambda)=\frac{1}{2} e^{-(x+\lambda)/2} \left (\frac{x}{\lambda}\right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})

where $I_\nu(y)$ is a modified Bessel function of the first kind given by

I_\nu(y) = (y/2)^\nu \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(\nu+j+1)} .

Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:^[1]

f_X(x;k,\lambda)={{\rm e}^{-\lambda/2}} _0F_1(;k/2;\lambda x/4)\frac{1}{2^{k/2}\Gamma(k/2)} {\rm e}^{-x/2} x^{k/2-1}.

Siegel (1979) discusses the case k = 0 specifically (zero degrees of freedom), in which case the distribution has a discrete component at zero.

Properties

Moment generating function

The moment generating function is given by

M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}.

Moments

The first few raw moments are:

\mu'_1=k+\lambda

\mu'_2=(k+\lambda)^2 + 2(k + 2\lambda)

\mu'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda)

\mu'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda)

The first few central moments are:

\mu_2=2(k+2\lambda)\,

\mu_3=8(k+3\lambda)\,

\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\,

The nth cumulant is

K_n=2^{n-1}(n-1)!(k+n\lambda).\,

Hence

\mu'_n = 2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1} \frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\lambda )\mu'_{n-j}.

Cumulative distribution function

Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as

P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} Q(x; k+2j)

where $Q(x; k)\,$ is the cumulative distribution function of the central chi-squared distribution with k degrees of freedom which is given by

Q(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,

and where

\gamma(k,z)\,

is the lower incomplete Gamma function.

The Marcum Q-function $Q_M(a,b)$ can also be used to represent the cdf.^[2]

P(x; k, \lambda) = 1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)

Approximation

Sankaran ^[3] discusses a number of closed form approximations for the cumulative distribution function. In an earlier paper,^[4] he derived and states the following approximation:

P(x; k, \lambda ) \approx \Phi \left\{ \frac{(\frac{x} {k + \lambda}) ^ h - (1 + h p (h - 1 - 0.5 (2 - h) m p))} {h \sqrt{2p} (1 + 0.5 m p)} \right\}

where

\Phi \lbrace \cdot \rbrace \,

denotes the cumulative distribution function of the standard normal distribution;

h = 1 - \frac{2}{3} \frac{(k+ \lambda) (k+ 3 \lambda)}{(k+ 2 \lambda) ^ 2} \, ;

p = \frac{k+ 2 \lambda}{(k+ \lambda) ^ 2} ;

m = (h - 1) (1 - 3 h) \, .

This and other approximations are discussed in a later text book.^[5]

To approximate the chi-squared distribution, the non-centrality parameter, $\lambda\,$ , is set to zero, yielding

P(x; k, \lambda ) \approx \Phi \left\{ \frac{\left(\frac{x}{k}\right)^{1/3} - \left(1 - \frac{2}{9k}\right) } {\sqrt{\frac{2}{9k}} } \right\} ,

essentially approximating the normalized chi-squared distribution X / k as the cube of a Gaussian.

For a given probability, the formula is easily inverted to provide the corresponding approximation for $x$ .

Differential equation

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

\left\{\begin{array}{l} 4 x f''(x)+(-2 k+4 x+8) f'(x)+f(x) (-k-\lambda+x+4)=0 \\[10pt] f(1) 2^{k/2} e^{\frac{\lambda+1}{2}}=\, _0\tilde{F}_1\left(;\frac{k}{2};\frac{\lambda}{4}\right) \\[10pt] \lambda \, _0\tilde{F}_1\left(;\frac{k}{2}+1;\frac{\lambda}{4}\right)+2 (k-3) \, _0\tilde{F}_1\left(;\frac{k}{2};\frac{\lambda}{4}\right)= 2^{\frac{k}{2}+2} e^{\frac{\lambda +1}{2}} f'(1) \end{array}\right\}

Derivation of the pdf

The derivation of the probability density function is most easily done by performing the following steps:

First, assume without loss of generality that $\sigma_1=\cdots=\sigma_k=1$ . Then the joint distribution of $X_1,\ldots,X_k$ is spherically symmetric, up to a location shift.
The spherical symmetry then implies that the distribution of $X=X_1^2+\cdots+X_k^2$ depends on the means only through the squared length, $\lambda=\mu_1^2+\cdots+\mu_k^2$ . Without loss of generality, we can therefore take $\mu_1=\sqrt{\lambda}$ and $\mu_2=\cdots=\mu_k=0$ .
Now derive the density of $X=X_1^2$ (i.e. the k = 1 case). Simple transformation of random variables shows that

\begin{align}f_X(x,1,\lambda) &= \frac{1}{2\sqrt{x}}\left( \phi(\sqrt{x}-\sqrt{\lambda}) + \phi(\sqrt{x}+\sqrt{\lambda}) \right )\\ &= \frac{1}{\sqrt{2\pi x}} e^{-(x+\lambda)/2} \cosh(\sqrt{\lambda x}), \end{align}

where

\phi(\cdot)

is the standard normal density.

Expand the cosh term in a Taylor series. This gives the Poisson-weighted mixture representation of the density, still for k = 1. The indices on the chi-squared random variables in the series above are 1 + 2i in this case.
Finally, for the general case. We've assumed, without loss of generality, that $X_2,\ldots,X_k$ are standard normal, and so $X_2^2+\cdots+X_k^2$ has a central chi-squared distribution with (k − 1) degrees of freedom, independent of $X_1^2$ . Using the poisson-weighted mixture representation for $X_1^2$ , and the fact that the sum of chi-squared random variables is also chi-squared, completes the result. The indices in the series are (1 + 2i) + (k − 1) = k + 2i as required.

Related distributions

If $V$ is chi-squared distributed $V \sim \chi_k^2$ then $V$ is also non-central chi-squared distributed: $V \sim {\chi'}^2_k(0)$
If $V_1 \sim {\chi'}_{k_1}^2(\lambda)$ and $V_2 \sim {\chi'}_{k_2}^2(0)$ and $V_1$ is independent of $V_2$ then a noncentral F-distributed variable is developed as $\frac{V_1/k_1}{V_2/k_2} \sim F'_{k_1,k_2}(\lambda)$
If $J \sim \mathrm{Poisson}\left({\lambda}\right)$ , then $\chi_{k+2J}^2 \sim {\chi'}_k^2(\lambda)$
If $V\sim{\chi'}^2_2(\lambda)$ , then $\sqrt{V}$ takes the Rice distribution with parameter $\sqrt{\lambda}$ .
Normal approximation:^[6] if $V \sim {\chi'}^2_k(\lambda)$ , then $\frac{V-(k+\lambda)}{\sqrt{2(k+2\lambda)}}\to N(0,1)$ in distribution as either $k\to\infty$ or $\lambda\to\infty$ .

Transformations

Sankaran (1963) discusses the transformations of the form $z=[(X-b)/(k+\lambda)]^{1/2}$ . He analyzes the expansions of the cumulants of $z$ up to the term $O((k+\lambda)^{-4})$ and shows that the following choices of $b$ produce reasonable results:

$b=(k-1)/2$ makes the second cumulant of $z$ approximately independent of $\lambda$
$b=(k-1)/3$ makes the third cumulant of $z$ approximately independent of $\lambda$
$b=(k-1)/4$ makes the fourth cumulant of $z$ approximately independent of $\lambda$

Also, a simpler transformation $z_1 = (X-(k-1)/2)^{1/2}$ can be used as a variance stabilizing transformation that produces a random variable with mean $(\lambda + (k-1)/2)^{1/2}$ and variance $O((k+\lambda)^{-2})$ .

Usability of these transformations may be hampered by the need to take the square roots of negative numbers.

**Various chi and chi-squared distributions**
Name	Statistic
chi-squared distribution	$\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2$
noncentral chi-squared distribution	$\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2$
chi distribution	$\sqrt{\sum_1^k \left(\frac{X_i-\mu_i}{\sigma_i}\right)^2}$
noncentral chi distribution	$\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}$

Occurrences

Use in tolerance intervals

Two-sided normal regression tolerance intervals can be obtained based on the noncentral chi-squared distribution.^[7] This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.

Notes

↑ Muirhead (2005) Theorem 1.3.4
↑ Nuttall, Albert H. (1975): Some Integrals Involving the Q_M Function, IEEE Transactions on Information Theory, 21(1), 95–96, ISSN 0018-9448
↑ Sankaran , M. (1963). Approximations to the non-central chi-squared distribution Biometrika, 50(1-2), 199–204
↑ Sankaran , M. (1959). "On the non-central chi-squared distribution", Biometrika 46, 235–237
↑ Johnson et al. (1995) Section 29.8
↑ Muirhead (2005) pages 22–24 and problem 1.18.
↑ Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software 36 (5): 1–39. ISSN 1548-7660. Retrieved 19 February 2013. , p.32

References

Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
Johnson, N. L., Kotz, S., Balakrishnan, N. (1970), Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0-471-58494-0
Muirhead, R. (2005) Aspects of Multivariate Statistical Theory (2nd Edition). Wiley. ISBN 0-471-76985-1
Siegel, A.F. (1979), "The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity", Biometrika, 66, 381–386
Press, S.J. (1966), "Linear combinations of non-central chi-squared variates", The Annals of Mathematical Statistics 37 (2): 480–487, doi:10.1214/aoms/1177699531, JSTOR 2238621

External links

Non central chi squared distribution – from itfeature.com

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 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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