Noncentral chi-squared
Probability density function
Cumulative distribution function
Parameters	${\displaystyle k>0}$ degrees of freedom ${\displaystyle \lambda >0}$ non-centrality parameter
Support	${\displaystyle x\in [0,+\mathrm{\infty })}$
PDF	${\displaystyle \frac{1}{2}{e}^{-(x+\lambda )/2}{\left(\frac{x}{\lambda }\right)}^{k/4-1/2}{I}_{k/2-1}(\sqrt{\lambda x})}$
CDF	${\displaystyle 1-Q_{\frac{k}{2}}\left(\sqrt{\lambda },\sqrt{x}\right)}$ withMarcum Q-function${\displaystyle Q_M(a,b)}$
Mean	${\displaystyle k+\lambda }$
Variance	${\displaystyle 2(k+2\lambda )}$
Skewness	${\displaystyle \frac{2^{3/2}(k+3\lambda )}{(k+2\lambda {)}^{3/2}}}$
Excess kurtosis	${\displaystyle \frac{12(k+4\lambda )}{(k+2\lambda {)}^2}}$
MGF
CF	${\displaystyle \frac{\exp \left(\frac{i\lambda t}{1-2it}\right)}{(1-2it{)}^{k/2}}}$

Inprobability theory andstatistics, thenoncentral chi-squared distribution (or noncentral chi-square distribution,noncentral${\displaystyle {\chi }^2}$ distribution) is anoncentral generalization of thechi-squared distribution. It often arises in thepower analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chi-squared distribution; important examples of such tests are thelikelihood-ratio tests.^[1]

Let${\displaystyle (X_1,X_2,\dots ,X_i,\dots ,X_k)}$ bekindependent,normally distributed random variables with means${\displaystyle {\mu }_i}$ and unit variances. Then the random variable

${\displaystyle \sum _{i=1}^k{X}_i^2}$

is distributed according to the noncentral chi-squared distribution. It has two parameters:${\displaystyle k}$ which specifies the number ofdegrees of freedom (i.e. the number of${\displaystyle X_i}$), and${\displaystyle \lambda }$ which is related to the mean of the random variables${\displaystyle X_i}$ by:

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

${\displaystyle \lambda }$ is sometimes called thenoncentrality parameter. Note that some references define${\displaystyle \lambda }$ in other ways, such as half of the above sum, or its square root.

This distribution arises inmultivariate statistics as a derivative of themultivariate normal distribution. While the centralchi-squared distribution is the squarednorm of arandom vector with${\displaystyle N(0_k,I_k)}$ distribution (i.e., the squared distance from the origin to a point taken at random from that distribution), the non-central${\displaystyle {\chi }^2}$ is the squared norm of a random vector with${\displaystyle N(\mu ,I_k)}$ distribution. Here${\displaystyle 0_k}$ is a zero vector of lengthk,${\displaystyle \mu =({\mu }_1,\dots ,{\mu }_k)}$ and${\displaystyle I_k}$ is theidentity matrix of sizek.

Theprobability density function (pdf) is given by

${\displaystyle f_X(x;k,\lambda )=\sum _{i=0}^{\mathrm{\infty }}\frac{e^{-\lambda /2}(\lambda /2{)}^i}{i!}{f}_{Y_{k+2i}}(x),}$

where${\displaystyle Y_q}$ is distributed as chi-squared with${\displaystyle q}$ degrees of freedom.

From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weightedmixture of central chi-squared distributions. Suppose that a random variableJ has aPoisson distribution with mean${\displaystyle \lambda /2}$, and theconditional distribution ofZ givenJ = i is chi-squared withk + 2i degrees of freedom. Then theunconditional distribution ofZ is non-central chi-squared withk degrees of freedom, and non-centrality parameter${\displaystyle \lambda }$.

Alternatively, the pdf can be written as

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

where${\displaystyle I_{\nu }(y)}$ is a modifiedBessel function of the first kind given by

${\displaystyle I_{\nu }(y)=(y/2{)}^{\nu }\sum _{j=0}^{\mathrm{\infty }}\frac{(y^2/4{)}^j}{j!\mathrm{\Gamma }(\nu +j+1)}.}$

Using the relation betweenBessel functions andhypergeometric functions, the pdf can also be written as:^[2]

${\displaystyle f_X(x;k,\lambda )={{\mathrm{e}}^{-\lambda /2}}_0{F}_1(;k/2;\lambda x/4)\frac{1}{2^{k/2}\mathrm{\Gamma }(k/2)}{\mathrm{e}}^{-x/2}{x}^{k/2-1}.}$

The casek = 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).^[3][4]

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

1. Since${\displaystyle X_1,\dots ,X_k}$ have unit variances, their joint distribution is spherically symmetric, up to a location shift.
2. The spherical symmetry then implies that the distribution of${\displaystyle X=X_1^2+\cdots +X_k^2}$ depends on the means only through the squared length,${\displaystyle \lambda ={\mu }_1^2+\cdots +{\mu }_k^2}$. Without loss of generality, we can therefore take${\displaystyle {\mu }_1=\sqrt{\lambda }}$ and${\displaystyle {\mu }_2=\cdots ={\mu }_k=0}$.
3. Now derive the density of${\displaystyle X=X_1^2}$ (i.e. thek = 1 case). Simple transformation of random variables shows that

where${\displaystyle \varphi (\cdot )}$ is the standard normal density.

1. Expand thecosh term in aTaylor series. This gives the Poisson-weighted mixture representation of the density, still fork = 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${\displaystyle X_2,\dots ,X_k}$ are standard normal, and so${\displaystyle X_2^2+\cdots +X_k^2}$ has acentral chi-squared distribution with (k − 1) degrees of freedom, independent of${\displaystyle X_1^2}$. Using the poisson-weighted mixture representation for${\displaystyle X_1^2}$, 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.

Themoment-generating function is given by

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

The first few rawmoments are:

${\displaystyle {\mu }_1^{\prime }=k+\lambda }$

${\displaystyle {\mu }_2^{\prime }=(k+\lambda {)}^2+2(k+2\lambda )}$

${\displaystyle {\mu }_3^{\prime }=(k+\lambda {)}^3+6(k+\lambda )(k+2\lambda )+8(k+3\lambda )}$

${\displaystyle {\mu }_4^{\prime }=(k+\lambda {)}^4+12(k+\lambda {)}^2(k+2\lambda )+4(11k^2+44k\lambda +36{\lambda }^2)+48(k+4\lambda ).}$

The first few centralmoments are:

${\displaystyle {\mu }_2=2(k+2\lambda )}$

${\displaystyle {\mu }_3=8(k+3\lambda )}$

${\displaystyle {\mu }_4=12(k+2\lambda {)}^2+48(k+4\lambda )}$

Thenthcumulant is

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

Hence

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

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

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

where${\displaystyle Q(x;k)}$ is the cumulative distribution function of the central chi-squared distribution withk degrees of freedom which is given by

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

and where${\displaystyle \gamma (k,z)}$ is thelower incomplete gamma function.

TheMarcum Q-function${\displaystyle Q_M(a,b)}$ can also be used to represent the cdf.^[5]

${\displaystyle P(x;k,\lambda )=1-Q_{\frac{k}{2}}\left(\sqrt{\lambda },\sqrt{x}\right)}$

When the degrees of freedomk is positive odd integer, we have a closed form expression for the complementary cumulative distribution function given by^[6]

wheren is non-negative integer,Q is theGaussian Q-function, andI 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 ofhyperbolic functions.

In particular, fork = 1, we have

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

Also, fork = 3, we have

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

Abdel-Aty derives (as "first approx.") a non-centralWilson–Hilferty transformation:^[7]

${\displaystyle {\left(\frac{{\chi }^{\prime 2}}{k+\lambda }\right)}^{\frac{1}{3}}}$ is approximatelynormally distributed,${\displaystyle \sim \mathcal{N}\left(1-\frac{2}{9f},\frac{2}{9f}\right),}$ i.e.,

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

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

Sankaran discusses a number ofclosed formapproximations for thecumulative distribution function.^[8] In an earlier paper, he derived and states the following approximation:^[9]

${\displaystyle P(x;k,\lambda )\approx \mathrm{\Phi }\left\{\frac{(\frac{x}{k+\lambda }{)}^h-(1+hp(h-1-0.5(2-h)mp))}{h\sqrt{2p}(1+0.5mp)}\right\}}$

where

${\displaystyle \mathrm{\Phi }\{\cdot \}}$ denotes thecumulative distribution function of thestandard normal distribution;

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

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

${\displaystyle m=(h-1)(1-3h).}$

This and other approximations are discussed in a later text book.^[10]

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

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

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

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

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

• If${\displaystyle V}$ ischi-square distributed,${\displaystyle V\sim {\chi }_k^2}$, then${\displaystyle V}$ is also non-central chi-square distributed:${\displaystyle V\sim {{\chi }^{\prime }}_k^2(0)}$
• A linear combination of independent noncentral chi-squared variables${\displaystyle \xi =\sum _i{\lambda }_i{Y}_i+c,Y_i\sim {\chi }^{\prime 2}(m_i,{\delta }_i^2)}$, isgeneralized chi-square distributed.
• If${\displaystyle V_1\sim {{\chi }^{\prime }}_{k_1}^2(\lambda )}$ and${\displaystyle V_2\sim {{\chi }^{\prime }}_{k_2}^2(0)}$ and${\displaystyle V_1}$ is independent of${\displaystyle V_2}$ then anoncentralF-distributed variable is developed as${\displaystyle \frac{V_1/k_1}{V_2/k_2}\sim F_{k_1,k_2}^{\prime }(\lambda )}$
• If${\displaystyle J\sim \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\left(\frac{1}{2}\lambda \right)}$, then${\displaystyle {\chi }_{k+2J}^2\sim {{\chi }^{\prime }}_k^2(\lambda )}$
• If${\displaystyle V\sim {{\chi }^{\prime }}_2^2(\lambda )}$, then${\displaystyle \sqrt{V}}$ takes theRice distribution with parameter${\displaystyle \sqrt{\lambda }}$.
• Normal approximation:^[11] if${\displaystyle V\sim {{\chi }^{\prime }}_k^2(\lambda )}$, then${\displaystyle \frac{V-(k+\lambda )}{\sqrt{2(k+2\lambda )}}\to N(0,1)}$ in distribution as either${\displaystyle k\to \mathrm{\infty }}$ or${\displaystyle \lambda \to \mathrm{\infty }}$.
• If${\displaystyle V_1\sim {{\chi }^{\prime }}_{k_1}^2({\lambda }_1)}$and${\displaystyle V_2\sim {{\chi }^{\prime }}_{k_2}^2({\lambda }_2)}$, where${\displaystyle V_1,V_2}$ are independent, then${\displaystyle W=(V_1+V_2)\sim {{\chi }^{\prime }}_{k_1+k_2}^2({\lambda }_1+{\lambda }_2)}$.
• In general, for an independent finite set of${\displaystyle V_i\sim {{\chi }^{\prime }}_{k_i}^2({\lambda }_i)}$,${\displaystyle i\in \{1,\dots ,N\}}$, the sum of these non-central chi-square distributed random variables${\displaystyle Y=\sum _{i=1}^N{V}_i}$ has the distribution${\displaystyle Y\sim {{\chi }^{\prime }}_{k_y}^2({\lambda }_y)}$ where${\displaystyle k_y=\sum _{i=1}^N{k}_i}$,${\displaystyle {\lambda }_y=\sum _{i=1}^N{\lambda }_i}$. This can be seen using moment generating functions as follows:${\displaystyle M_Y(t)=M_{\sum _{i=1}^N{V}_i}(t)=\prod _{i=1}^N{M}_{V_i}(t)}$ by the independence of the${\displaystyle V_i}$ 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.
• Thecomplex noncentral chi-squared distribution has applications in radio communication and radar systems.^{[citation needed]} Let${\displaystyle z_1,\dots ,z_k}$ be independent scalarcomplex random variables with noncentral circular symmetry, means of${\displaystyle {\mu }_i}$ and unit variances:${\displaystyle \mathrm{E}{\left|z_i-{\mu }_i\right|}^2=1}$. Then the real random variable${\displaystyle S=\sum _{i=1}^k{\left|z_i\right|}^2}$ is distributed according to the complex noncentral chi-squared distribution, which is effectively a scaled (by 1/2) non-central${\displaystyle {{\chi }^{\prime }}^2}$ with twice the degrees of freedom and twice the noncentrality parameter:

${\displaystyle f_S(S)={\left(\frac{S}{\lambda }\right)}^{(k-1)/2}{e}^{-(S+\lambda )}{I}_{k-1}(2\sqrt{S\lambda })}$,

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

Sankaran (1963) discusses the transformations of the form${\displaystyle z=[(X-b)/(k+\lambda ){]}^{1/2}}$. He analyzes the expansions of thecumulants of${\displaystyle z}$ up to the term${\displaystyle O((k+\lambda {)}^{-4})}$ and shows that the following choices of${\displaystyle b}$ produce reasonable results:

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

Also, a simpler transformation${\displaystyle z_1=(X-(k-1)/2{)}^{1/2}}$ can be used as avariance stabilizing transformation that produces a random variable with mean${\displaystyle (\lambda +(k-1)/2{)}^{1/2}}$ and variance${\displaystyle O((k+\lambda {)}^{-2})}$.

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	${\displaystyle \sum _{i=1}^k{\left(\frac{X_i-{\mu }_i}{{\sigma }_i}\right)}^2}$
noncentral chi-squared distribution	${\displaystyle \sum _{i=1}^k{\left(\frac{X_i}{{\sigma }_i}\right)}^2}$
chi distribution	${\displaystyle \sqrt{\sum _{i=1}^k{\left(\frac{X_i-{\mu }_i}{{\sigma }_i}\right)}^2}}$
noncentral chi distribution	${\displaystyle \sqrt{\sum _{i=1}^k{\left(\frac{X_i}{{\sigma }_i}\right)}^2}}$

Two-sided normalregressiontolerance intervals can be obtained based on the noncentral chi-squared distribution.^[12] 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
• Press, S.J. (1966), "Linear combinations of non-central chi-squared variates",The Annals of Mathematical Statistics,37 (2):480–487,doi:10.1214/aoms/1177699531,JSTOR 2238621

• Continuous distributions
• Noncentral distributions

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