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	$x \in [0; +\infty)\,$
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

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 pdf

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

1. Start with the joint PDF of two independent non-zero mean Gaussian distributions, $X$ and $Y$.
2. Convert the joint density $f(X, Y)$ to polar: $f(R, A)$ where $R^2 = (X^2+Y^2)$, $tan(A) = Y/X$.
3. Integrate over the angular variable $A$.
4. 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 $I_0$.
5. Take the Laplace (Fourier) transform term-by-term and the special case K = 2 and the MGF will result.
6. For the general case, take the K = 2 MGF and raise it to the $K/2$ power.
7. 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 distributions

• If $X$ is chi-square distributed $X\sim\chi^2_k$ 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	$\sqrt{\sum_1^k \left(\frac{X_i}{\sigma_i}\right)^2}$

References

• 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.

Related Links

This distribution (and many others) is available in the free interactive statistical tables program, STATTAB. The cumulative distribution function, its inverse, and parameters of the distribution can be calculated with these packages. A free Fortran library for these distributions is in CDFLIB. The URL for download is [1]