Wikia

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory

**Noncentral chi-square**
Probability density function
Cumulative distribution function
Parameters	$k > 0\,$ degrees of freedom $\lambda > 0\,$ non-centrality parameter
Support	$x \in [0; +\infty)\,$
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	: $\sum_{j=0}^\infty e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} \frac{\gamma(j+k/2,x/2)}{\Gamma(j+k/2)}\,$
Mean	$k+\lambda\,$
Median
Mode
Variance	$2(k+2\lambda)\,$
Skewness	$\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}$
Kurtosis	$\frac{12(k+4\lambda)}{(k+2\lambda)^2}$
Entropy
mgf	$\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}$
Char. func.	$\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}$

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

$\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2$

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 $\lambda$ which is related to the mean of the random variables $X_i$ by:

$\lambda=\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2$ .

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

Properties Edit

The noncentral chi-square distribution is equivalent to a (central) chi-square distribution with $k + 2P$ degrees of freedom, where $P$ is a Poisson random variable with parameter $\lambda/2$ . Thus, the probability distribution 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.

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 $I_\nu(z)$ 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 density of the central chi-squared distribution which is given by

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

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

Derivation of the pdfEdit

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

Start with the joint PDF of two independent non-zero mean Gaussian distributions, $X$ and $Y$ .
Convert the joint density $f(X, Y)$ to polar: $f(R, A)$ where $R^2 = (X^2+Y^2)$ , $tan(A) = Y/X$ .
Integrate over the angular variable $A$ .
Convert from R to r where $r^2 = R$ . This will yield a series expansion in r one factor of which matches the modified Bessel function $I_0$ .
Take the Laplace (Fourier) transform term-by-term and the special case K = 2 and the MGF will result.
For the general case, take the K = 2 MGF and raise it to the $K/2$ power.
The final trick to hide the K-dependence in the numerator of the MGF is to note that $\lambda$ is a function of K; that is,

$\lambda_2=\sum_1^2 \left(\frac{\mu_i}{\sigma_i}\right)^2$

$\lambda_K=\sum_1^k \left(\frac{\mu_i}{\sigma_i}\right)^2 = \lambda$

and therefore, $\lambda$ is not explicitly a function of K in the above table.

Related distributionsEdit

If $X$ is chi-square distributed $X \sim \chi_k^2$ then $X^2$ is also non-central chi-square distributed: $X^2 \sim NC\chi^2_k(0)$

If $J \sim Poisson(\lambda)$ , then $\chi_k^2(\lambda) \sim \chi_{k+2J}^2$

**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	$\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}$

	Probability distributions [[[:Template:Tnavbar-plain-nodiv]]]
	Univariate	Multivariate
Discrete:	Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • degree • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle	Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling

</center>

ReferencesEdit

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.

Wikia

Noncentral chi-square distribution

Contents

Properties Edit

Derivation of the pdfEdit

Related distributionsEdit

ReferencesEdit

Related Links Edit

Photos

Recent Wiki Activity