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	$x \in [0; %2B\infty)\,$
PDF	$\frac{1}{2}e^{-(x%2B\lambda)/2}\left (\frac{x}{\lambda} \right)^{k/4-1/2} I_{k/2-1}(\sqrt{\lambda x})$
CDF	$1 - Q_{\frac{k}{2}} \left( \sqrt{\lambda}, \sqrt{x} \right)$ with Marcum Q-function $Q_M(a,b)$
Mean	$k%2B\lambda\,$
Variance	$2(k%2B2\lambda)\,$
Skewness	$\frac{2^{3/2}(k%2B3\lambda)}{(k%2B2\lambda)^{3/2}}$
Ex. kurtosis	$\frac{12(k%2B4\lambda)}{(k%2B2\lambda)^2}$
MGF	$\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}$ 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.

1 Background
2 Properties
3 Derivation of the pdf
4 Related distributions
- 4.1 Transformations
5 Notes
6 References

Background

Let $X_i$ be k independent, normally distributed random variables with means $\mu_i$ 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-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:

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

$\lambda$ is sometime 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, $\mu = (\mu_1, ..., \mu_k)$ and $I_k$ is the identity matrix of size k.

Properties

Probability density function

The probability density function is given by

$f_X(x; k,\lambda) = \sum_{i=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^i}{i!} f_{Y_{k%2B2i}}(x),$

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

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

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

$I_a(y) = (y/2)^a \sum_{j=0}^\infty \frac{ (y^2/4)^j}{j! \Gamma(a%2Bj%2B1)} .$

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.

Moment generating function

The moment generating function is given by

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

The first few raw moments are:

$\mu^'_1=k%2B\lambda$

$\mu^'_2=(k%2B\lambda)^2 %2B 2(k %2B 2\lambda)$

$\mu^'_3=(k%2B\lambda)^3 %2B 6(k%2B\lambda)(k%2B2\lambda)%2B8(k%2B3\lambda)$

$\mu^'_4=(k%2B\lambda)^4%2B12(k%2B\lambda)^2(k%2B2\lambda)%2B4(11k^2%2B44k\lambda%2B36\lambda^2)%2B48(k%2B4\lambda)$

The first few central moments are:

$\mu_2=2(k%2B2\lambda)\,$

$\mu_3=8(k%2B3\lambda)\,$

$\mu_4=12(k%2B2\lambda)^2%2B48(k%2B4\lambda)\,$

The nth cumulant is

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

Hence

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

Cumulative distribution function

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

$P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} Q(x; k%2B2j)$

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:

$P(x; k, \lambda ) \approx \Phi \lbrace \frac{(\frac{x} {k %2B \lambda}) ^ h - (1 %2B h p (h - 1 - 0.5 (2 - h) m p))} {h \sqrt{ 2p} (1 %2B 0.5 m p)} \rbrace$

where

$\Phi \lbrace \cdot \rbrace \,$ denotes the cumulative distribution function of the standard normal distribution;

$h = 1 - \frac{2}{3} \frac{(k%2B \lambda) (k%2B 3 \lambda)}{(k%2B 2 \lambda) ^ 2} \,�;$

$p = \frac{k%2B 2 \lambda}{(k%2B \lambda) ^ 2}�;$

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

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.

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

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 $X=X_1^2%2B\ldots%2BX_k^2$ depends on the means only through the squared length, $\lambda=\mu_1^2%2B\ldots%2B\mu_k^2$ . 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 : $\begin{align}f_X(x,1,\lambda) &= \frac{1}{2\sqrt{x}}\left( \phi(\sqrt{x}-\sqrt{\lambda}) %2B \phi(\sqrt{x}%2B\sqrt{\lambda}) \right )\\ &= \frac{1}{\sqrt{2\pi x}} e^{-(x%2B\lambda)/2} \cosh(\sqrt{\lambda x}),\\ \end{align}$
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 $X_2^2%2B\ldots%2BX_k^2$ 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 $J \sim Poisson(\frac{\lambda}{2})$ , then $\chi_{k%2B2J}^2 \sim {\chi'}_k^2(\lambda)$

Normal approximation^[6]: if $V \sim {\chi'}^2_k(\lambda)$ , then $\frac{V-(k%2B\lambda)}{\sqrt{2(k%2B2\lambda)}}\to N(0,1)$ in distribution as either $k\to\infty$ or $\lambda\to\infty$ .

Transformations

Sankaran (1963) discusses the transformations of the form $z=[(X-b)/(k%2B\lambda)]^{1/2}$ . He analyzes the expansions of the cumulants of $z$ up to the term $O((k%2B\lambda)^{-4})$ 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 $(\lambda %2B (k-1)/2)^{1/2}$ and variance $O((k%2B\lambda)^{-2})$ .

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

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.

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, Wiley
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, JSTOR 2238621

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