Noncentral chi-square distribution

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Noncentral chi-square
Probability density function
Cumulative distribution function
Parameters	$k > 0\,$ degrees of freedom $\lambda > 0\,$ non-centrality parameter
Support	$x \in [0; +\infty)\,$
Probability density function (pdf)	$\frac{1}{2}e^{-(x+\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$
Cumulative distribution function (cdf)	: $\sum_{j=0}^\infty e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} \frac{\gamma(j+k/2,x/2)}{\Gamma(j+k/2)}\,$
Mean	$k+\lambda\,$
Median
Mode
Variance	$2(k+2\lambda)\,$
Skewness	$\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}$
Excess kurtosis	$\frac{12(k+4\lambda)}{(k+2\lambda)^2}$
Entropy
Moment-generating function (mgf)	$\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}$ for $2 t < 1$
Characteristic function	$\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}$

In probability theory and statistics, the noncentral chi-square or noncentral $? 2$ distribution is a generalization of the chi-square distribution. If $X i$ are k independent, normally distributed random variables with means $µ i$ and variances $\sigma_i^2$ , then the random variable

$\sum_{i=1}^k \left(\frac{X_i}{\sigma_i}\right)^2$

is distributed according to the noncentral chi-square distribution. The noncentral chi-square distribution has two parameters: $k$ which specifies the number of degrees of freedom (i.e. the number of $X i$ ), and $?$ which is related to the mean of the random variables $X i$ by:

$\lambda=\sum_{i=1}^k \left(\frac{\mu_i}{\sigma_i}\right)^2.$

Note that some references define $?$ as one half of the above sum.

1 Properties
2 Derivation of the pdf
3 Related distributions
4 Software and online calculator
5 References

Properties

The probability density function 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-square 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 $? / 2$ , and the conditional distribution of Z given $J = j$ 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 $?$ .

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 $I ? (z)$ is a modified Bessel function of the first kind given by

$I_a(y) := (y/2)^a \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(a+j+1)}$

The moment generating function is given by

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

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

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

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

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

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

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

where $?(k, z)$ is the lower incomplete Gamma function.

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 $\sigma_1=\ldots=\sigma_k=1$ . 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 $X=X_1^2+\ldots+X_k^2$ depends on the means only through the squared length, $\lambda=\mu_1^2+\ldots+\mu_k^2$ . Without loss of generality, we can therefore take $\mu_1=\sqrt{\lambda}$ and $\mu_2=\dots=\mu_k=0$ .
Now derive the density of $X=X_1^2$ (i.e. 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, wlog, that $X_2,\ldots,X_k$ are standard normal, and so $X_2^2+\ldots+X_k^2$ 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-square distributed $V \sim \chi_k^2$ then $V$ is also non-central chi-square distributed: $V \sim {\chi'}^2_k(0)$

If $V_1 \sim {\chi'}_{k_1}^2(\lambda)$ and $V_2 \sim {\chi'}_{k_2}^2(0)$ 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 $J \sim Poisson(\frac{\lambda}{2})$ , then $\chi_{k+2J}^2 \sim {\chi'}_k^2(\lambda)$

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

Software and online calculator

Many statistical software packages and libraries include functions for computing noncentral chisquare densities and probabilities. The table below gives commands for the following example problems:

Density $f X (x; k,?)$ with x=5.0, k=3, $? = 1.5$
Cumulative probability $P (x; k,?)$ with x=5.0, k=3, $? = 1.5$
Quantile: Find x in $P (x; k,?) = q$ with k=3, $? = 1.5$ , q=0.5
Critical noncentral parameter: Find $?$ in $P (x; k,?) = q$ with x=5.0,k=3, and q=0.5
Random numbers: Generate 100 random observations from the distribution with k=3, $?$ =1.5

Software	Density	Cumulative Prob.	Quantile	Noncentral parameter	Random numbers
Matlab	ncx2pdf(5.0,3,1.5)	ncx2cdf(5.0,3,1.5)	ncx2inv(.5,3,1.5)	fsolve(@(L)(ncx2cdf(5.0,3,L)-.5), 1)	ncx2rnd(3,1.5,100,1)
R	dchisq(5.0,3,1.5)	pchisq(5.0,3,1.5)	qchisq(.5,3,1.5)	require(MBESS); conf.limits.nc.chisq( 5,NULL,3,0,.5)	rchisq(100,3,1.5)
SAS	PROBCHI(5.0,3,1.5)	CDF('CHISQUARE',5.0,3,1.5)	CINV(.5,3,1.5)	CNONCT(5.0,3,.55)	?
Stattab ^[1]	NA	5.0 3 1.5 ? .	? 3 1.5 0.5 .	?	NA
Correct Answer	0.097257	0.649285	3.668745	2.898530	Varies

Note: Any software that produces the answers 0.101384, 0.490071, 5.09848 for the first three problems is including a factor of 0.5 in the definition of the noncentrality parameter. This is standard in statistics texts (e.g. ^[2]), but apparently not among programmers who don't read before writing their code.

These parameters can also be calculated online.

References

^ MD Anderson Cancer Center [1]
^ R. Christensen, Plane Answers to Complex Questions (3rd edition, 2002), Springer, NY, p.424.

Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions, Dover. Section 26.4.25.
Johnson, N. L. and Kotz, S., (1970), Continuous Univariate Distributions, vol. 2, Houghton-Mifflin.

v • d • e

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin-Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,8)

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-8,8)

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Category: Continuous distributions

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