noncentral chi-squared distribution

Let $(X\_1,X\_2,\; \backslash ldots,\; X\_i,\; \backslash ldots,X\_k)$ be k independent, normally distributed random variables with means $\backslash mu\_i$ and unit variances. Then the random variable

: $\backslash sum\_\{i=1\}^k\; X\_i^2$

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 $\backslash lambda$ which is related to the mean of the random variables $X\_i$ by:

: $\backslash lambda=\backslash sum\_\{i=1\}^k\; \backslash mu\_i^2.$

$\backslash lambda$ is sometimes called the noncentrality parameter. Note that some references define $\backslash lambda$ 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 to a point taken at random from that distribution), the non-central $\backslash chi^2$ is the squared norm of a random vector with $N(\backslash mu,I\_k)$ distribution. Here $0\_k$ is a zero vector of length k, $\backslash mu\; =\; (\backslash mu\_1,\; \backslash ldots,\; \backslash mu\_k)$ and $I\_k$ is the identity matrix of size k.

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 $\backslash lambda/2$, 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 $\backslash lambda$.

Alternatively, the pdf can be written as

: $f\_X(x;k,\backslash lambda)=\backslash frac\; 1\; 2\; e^\{-(x+\backslash lambda)/2\}\; \backslash left\; (\backslash frac\; x\; \backslash lambda\; \backslash right)^\{k/4-1/2\}\; I\_\{k/2-1\}(\backslash sqrt\{\backslash lambda\; x\})$

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

:$I\_\backslash nu(y)\; =\; (y/2)^\backslash nu\; \backslash sum\_\{j=0\}^\backslash infty\; \backslash frac\{\; (y^2/4)^j\}\{j!\; \backslash Gamma(\backslash nu+j+1)\}.$

Using the relation between Bessel functions and hypergeometric functions, the pdf can also be written as:Muirhead (2005) Theorem 1.3.4

:$f\_X(x;k,\backslash lambda)=\{\{\backslash rm\; e\}^\{-\backslash lambda/2\}\}\; \_0F\_1(;k/2;\backslash lambda\; x/4)\backslash frac\; 1\; \{2^\{k/2\}\; \backslash Gamma(k/2)\}\; \{\backslash rm\; e\}^\{-x/2\}\; x^\{k/2-1\}.$

The case k = 0 (zero degrees of freedom), in which case the distribution has a discrete component at zero, is discussed by Torgersen (1972) and further by Siegel (1979).Torgersen, E. N. (1972), "Supplementary notes on linear models", Preprint series: Statistical Memoirs, Dept. of Mathematics, University of Oslo, http://urn.nb.no/URN:NBN:no-58681Siegel, A. F. (1979), "The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity", Biometrika, 66, 381–386

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

1. Since $X\_1,\backslash ldots,X\_k$ have unit variances, their joint distribution is spherically symmetric, up to a location shift.
2. The spherical symmetry then implies that the distribution of $X=X\_1^2+\backslash cdots+X\_k^2$ depends on the means only through the squared length, $\backslash lambda=\backslash mu\_1^2+\backslash cdots+\backslash mu\_k^2$. Without loss of generality, we can therefore take $\backslash mu\_1=\backslash sqrt\{\backslash lambda\}$ and $\backslash mu\_2=\backslash cdots=\backslash mu\_k=0$.
3. Now derive the density of $X=X\_1^2$ (i.e. the k = 1 case). Simple transformation of random variables shows that

:::

::where $\backslash phi(\backslash cdot)$ is the standard normal density.

1. 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.
2. Finally, for the general case. We've assumed, without loss of generality, that $X\_2,\backslash ldots,X\_k$ are standard normal, and so $X\_2^2+\backslash cdots+X\_k^2$ 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 a chi-square, completes the result. The indices in the series are (1 + 2i) + (k − 1) = k + 2i as required.

The moment-generating function is given by

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

The first few raw moments are:

:$\backslash mu\text{'}\_1=k+\backslash lambda$

:$\backslash mu\text{'}\_2=(k+\backslash lambda)^2\; +\; 2(k\; +\; 2\backslash lambda)$

:$\backslash mu\text{'}\_3=(k+\backslash lambda)^3\; +\; 6(k+\backslash lambda)(k+2\backslash lambda)+8(k+3\backslash lambda)$

:$\backslash mu\text{'}\_4=(k+\backslash lambda)^4+12(k+\backslash lambda)^2(k+2\backslash lambda)+4(11k^2+44k\backslash lambda+36\backslash lambda^2)+48(k+4\backslash lambda).$

The first few central moments are:

:$\backslash mu\_2=2(k+2\backslash lambda)\backslash ,$

:$\backslash mu\_3=8(k+3\backslash lambda)\backslash ,$

:$\backslash mu\_4=12(k+2\backslash lambda)^2+48(k+4\backslash lambda)\backslash ,$

The nth cumulant is

:$\backslash kappa\_n=2^\{n-1\}(n-1)!(k+n\backslash lambda).\backslash ,$

Hence

:$\backslash mu\text{'}\_n\; =\; 2^\{n-1\}(n-1)!(k+n\backslash lambda)+\backslash sum\_\{j=1\}^\{n-1\}\; \backslash frac\{(n-1)!2^\{j-1\}\}\{(n-j)!\}(k+j\backslash lambda\; )\backslash mu\text{'}\_\{n-j\}.$

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

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

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

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

:and where $\backslash gamma(k,z)\backslash ,$ is the lower incomplete gamma function.

The Marcum Q-function $Q\_M(a,b)$ can also be used to represent the cdf.Nuttall, Albert H. (1975): [https://ieeexplore.ieee.org/document/1055327/;jsessionid=B83CF5D65D889B2C82555368D5AEF7F5?arnumber=1055327 Some Integrals Involving the Q_M Function], IEEE Transactions on Information Theory, 21(1), 95–96, {{ISSN|0018-9448}}

:$P(x;\; k,\; \backslash lambda)\; =\; 1\; -\; Q\_\{\backslash frac\{k\}\{2\}\}\; \backslash left(\; \backslash sqrt\{\backslash lambda\},\; \backslash sqrt\{x\}\; \backslash right)$

When the degrees of freedom k is positive odd integer, we have a closed form expression for the complementary cumulative distribution function given byA. Annamalai, C. Tellambura and John Matyjas (2009). "A New Twist on the Generalized Marcum Q-Function Q_M(a, b) with Fractional-Order M and its Applications". 2009 6th IEEE Consumer Communications and Networking Conference, 1–5, {{ISBN|978-1-4244-2308-8}}

:$$

\begin{align}

P(x; 2n+1, \lambda) &= 1 - Q_{n+1/2}(\sqrt{\lambda}, \sqrt{x}) \\

&= 1 - \left[ Q(\sqrt{x}-\sqrt{\lambda}) + Q(\sqrt{x}+\sqrt{\lambda}) + e^{-(x + \lambda)/2} \sum_{m=1}^n \left(\frac{x}{\lambda}\right)^{m/2-1/4} I_{m-1/2}(\sqrt{\lambda x}) \right],

\end{align}

where n is non-negative integer, Q is the Gaussian Q-function, and I is the modified Bessel function of first kind with half-integer order. The modified Bessel function of first kind with half-integer order in itself can be represented as a finite sum in terms of hyperbolic functions.

In particular, for k = 1, we have

:$P(x;\; 1,\; \backslash lambda)\; =\; 1\; -\; \backslash left[\; Q(\backslash sqrt\{x\}-\backslash sqrt\{\backslash lambda\})\; +\; Q(\backslash sqrt\{x\}+\backslash sqrt\{\backslash lambda\})\; \backslash right].$

Also, for k = 3, we have

:$P(x;\; 3,\; \backslash lambda)\; =\; 1\; -\; \backslash left[\; Q(\backslash sqrt\{x\}-\backslash sqrt\{\backslash lambda\})\; +\; Q(\backslash sqrt\{x\}+\backslash sqrt\{\backslash lambda\})\; +\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash frac\{\backslash sinh\; (\backslash sqrt\{\backslash lambda\; x\})\}\{\backslash sqrt\{\backslash lambda\}\}\; e^\{-(x+\backslash lambda)/2\}\; \backslash right].$

Abdel-Aty derives (as "first approx.") a non-central Wilson–Hilferty transformation:{{cite journal |last=Abdel-Aty |first=S. |year=1954 |jstor=2332731 |title=Approximate Formulae for the Percentage Points and the Probability Integral of the Non-Central χ² Distribution |journal=Biometrika |volume=41 |issue= 3/4|pages=538–540 |doi=10.2307/2332731 }}

$\backslash left(\backslash frac\{\backslash chi\text{'}^2\}\{k+\backslash lambda\}\backslash right)^\{\backslash frac\; 1\; 3\}$ is approximately normally distributed, $\backslash sim\; \backslash mathcal\{N\}\backslash left(1-\backslash frac\{2\}\{9f\},\; \backslash frac\{2\}\{9f\}\; \backslash right),$ i.e.,

:$P(x;\; k,\; \backslash lambda\; )\backslash approx\; \backslash Phi\; \backslash left\backslash \{\; \backslash frac\{\backslash left(\backslash frac\{x\}\{k+\backslash lambda\}\backslash right)^\{1/3\}\; -\; \backslash left(1\; -\; \backslash frac\{2\}\{9f\}\backslash right)\; \}\; \{\backslash sqrt\{\backslash frac\{2\}\{9f\}\}\; \}\; \backslash right\backslash \},\; \backslash text\{where\; \}\; \backslash \; f\; :=\; \backslash frac\{(k+\backslash lambda)^2\}\{k+2\backslash lambda\}\; =\; k\; +\; \backslash frac\{\backslash lambda^2\}\{k+2\backslash lambda\},$

which is quite accurate and well adapting to the noncentrality. Also, $f\; =\; f(k,\backslash lambda)$ becomes $f\; =\; k$ for $\backslash lambda=0$, the (central) chi-squared case.

Sankaran discusses a number of closed form approximations for the cumulative distribution function.{{cite journal |last=Sankaran |first=M. |year=1963 |doi=10.1093/biomet/50.1-2.199 |title=Approximations to the non-central chi-squared distribution |journal=Biometrika |volume=50 |issue=1–2 |pages=199–204 }} In an earlier paper, he derived and states the following approximation:{{cite journal |last=Sankaran |first=M. |year=1959 |title=On the non-central chi-squared distribution |journal=Biometrika |volume=46 |issue=1–2 |pages=235–237 |doi=10.1093/biomet/46.1-2.235 }}

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

where

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

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

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

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

This and other approximations are discussed in a later text book.Johnson et al. (1995) Continuous Univariate Distributions Section 29.8

More recently, since the CDF of non-central chi-squared distribution with odd degree of freedom can be exactly computed, the CDF for even degree of freedom can be approximated by exploiting the monotonicity and log-concavity properties of Marcum-Q function as

:$P(x;\; 2n,\; \backslash lambda\; )\; \backslash approx\; \backslash frac\{1\}\{2\}\backslash left[\; P(x;\; 2n\; -\; 1,\; \backslash lambda)\; +\; P(x;\; 2n\; +\; 1,\; \backslash lambda)\; \backslash right].$

Another approximation that also serves as an upper bound is given by

:$P(x;\; 2n,\; \backslash lambda\; )\; \backslash approx\; 1\; -\; \backslash left[\; (1-\; P(x;\; 2n\; -\; 1,\; \backslash lambda))\; (1\; -\; P(x;\; 2n\; +\; 1,\; \backslash lambda))\; \backslash right]^\{1/2\}.$

For a given probability, these formulas are easily inverted to provide the corresponding approximation for $x$, to compute approximate quantiles.

• If $V$ is chi-square distributed, $V\; \backslash sim\; \backslash chi\_k^2$, then $V$ is also non-central chi-square distributed: $V\; \backslash sim\; \{\backslash chi\text{'}\}^2\_k(0)$
• A linear combination of independent noncentral chi-squared variables $\backslash xi=\backslash sum\_i\; \backslash lambda\_i\; Y\_i\; +\; c,\; \backslash quad\; Y\_i\; \backslash sim\; \backslash chi\text{'}^2(m\_i,\backslash delta\_i^2)$, is generalized chi-square distributed.
• If $V\_1\; \backslash sim\; \{\backslash chi\text{'}\}\_\{k\_1\}^2(\backslash lambda)$ and $V\_2\; \backslash sim\; \{\backslash chi\text{'}\}\_\{k\_2\}^2(0)$ and $V\_1$ is independent of $V\_2$ then a noncentral F-distributed variable is developed as $\backslash frac\{V\_1/k\_1\}\{V\_2/k\_2\}\; \backslash sim\; F\text{'}\_\{k\_1,k\_2\}(\backslash lambda)$
• If $J\; \backslash sim\; \backslash mathrm\{Poisson\}\backslash left(\{\backslash frac\{1\}\{2\}\backslash lambda\}\backslash right)$, then $\backslash chi\_\{k+2J\}^2\; \backslash sim\; \{\backslash chi\text{'}\}\_k^2(\backslash lambda)$
• If $V\backslash sim\{\backslash chi\text{'}\}^2\_2(\backslash lambda)$, then $\backslash sqrt\{V\}$ takes the Rice distribution with parameter $\backslash sqrt\{\backslash lambda\}$.
• Normal approximation:Muirhead (2005) pages 22–24 and problem 1.18. if $V\; \backslash sim\; \{\backslash chi\text{'}\}^2\_k(\backslash lambda)$, then $\backslash frac\{V-(k+\backslash lambda)\}\{\backslash sqrt\{2(k+2\backslash lambda)\}\}\backslash to\; N(0,1)$ in distribution as either $k\backslash to\backslash infty$ or $\backslash lambda\backslash to\backslash infty$.
• If $V\_1\; \backslash sim\; \{\backslash chi\text{'}\}^2\_\{k\_1\}(\backslash lambda\_1)$and $V\_2\; \backslash sim\; \{\backslash chi\text{'}\}^2\_\{k\_2\}(\backslash lambda\_2)$, where $V\_1,\; V\_2$ are independent, then $W\; =\; (V\_1+V\_2)\; \backslash sim\; \{\backslash chi\text{'}\}^2\_\{k\_1\; +\; k\_2\}(\backslash lambda\_1+\backslash lambda\_2)$.
• In general, for an independent finite set of $V\_i\; \backslash sim\; \{\backslash chi\text{'}\}^2\_\{k\_i\}(\backslash lambda\_i)$, $i\backslash in\; \backslash \{\; 1,\; \backslash ldots,\; N\backslash \}$, the sum of these non-central chi-square distributed random variables $Y\; =\; \backslash sum\_\{i=1\}^N\; V\_i$ has the distribution $Y\; \backslash sim\; \{\backslash chi\text{'}\}^2\_\{k\_y\}(\backslash lambda\_y)$ where $k\_y=\backslash sum\_\{i=1\}^N\; k\_i$, $\backslash lambda\_y=\backslash sum\_\{i=1\}^N\backslash lambda\_i$. This can be seen using moment generating functions as follows: $M\_Y(t)\; =\; M\_\{\backslash sum\_\{i=1\}^N\; V\_i\}(t)\; =\; \backslash prod\_\{i=1\}^N\; M\_\{V\_i\}(t)$ by the independence of the $V\_i$ random variables. It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF – this is left as an exercise. Alternatively it can be seen via the interpretation in the background section above as sums of squares of independent normally distributed random variables with variances of 1 and the specified means.
• The complex noncentral chi-squared distribution has applications in radio communication and radar systems.{{citation needed|date=September 2020}} Let $z\_1,\; \backslash ldots,\; z\_k$ be independent scalar complex random variables with noncentral circular symmetry, means of $\backslash mu\_i$ and unit variances: $\backslash operatorname\{E\}\; \backslash left\; |z\_i\; -\; \backslash mu\_i\; \backslash right\; |^2\; =\; 1$. Then the real random variable $S\; =\; \backslash sum\_\{i=1\}^k\; \backslash left\; |z\_i\; \backslash right\; |\; ^2$ is distributed according to the complex noncentral chi-squared distribution, which is effectively a scaled (by 1/2) non-central $\{\backslash chi\text{'}\}^2$ with twice the degrees of freedom and twice the noncentrality parameter:

:: $f\_S(S)\; =\; \backslash left(\; \backslash frac\; \{S\}\{\backslash lambda\}\; \backslash right)^\{(k-1)/2\}\; e^\{-(S\; +\; \backslash lambda)\; \}$

I_{k-1} (2 \sqrt {S\lambda} ),

:where $\backslash lambda=\backslash sum\_\{i=1\}^k\; \backslash left\; |\backslash mu\_i\; \backslash right\; |^2$.

Sankaran (1963) discusses the transformations of the form

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

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

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 $(\backslash lambda\; +\; (k-1)/2)^\{1/2\}$ and variance $O((k+\backslash lambda)^\{-2\})$.

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

Two-sided normal regression tolerance intervals can be obtained based on the noncentral chi-squared distribution.{{cite journal| author=Derek S. Young| title=tolerance: An R Package for Estimating Tolerance Intervals| journal=Journal of Statistical Software|date=August 2010| volume=36| number=5| pages=1–39| issn=1548-7660| url=http://www.jstatsoft.org/v36/i05| access-date=19 February 2013}}, p. 32 This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.

• Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover.
• Johnson, N. L., Kotz, S., Balakrishnan, N. (1995), Continuous Univariate Distributions, Volume 2 (2nd Edition), Wiley. {{isbn|0-471-58494-0}}
• Muirhead, R. (2005) Aspects of Multivariate Statistical Theory (2nd Edition). Wiley. {{isbn|0-471-76985-1}}
• {{Citation

| title = Linear combinations of non-central chi-squared variates

| jstor = 2238621

| year = 1966

| author = Press, S.J.

| journal = The Annals of Mathematical Statistics

| pages = 480–487

| volume = 37

| issue = 2

| doi=10.1214/aoms/1177699531| doi-access = free

}}

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

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