Generalized chi-squared distribution
Probability density function
Cumulative distribution function
Notation	${\displaystyle \tilde{\chi }(\mathit{w},\mathit{k},\mathit{\lambda },s,m)}$
Parameters	${\displaystyle \mathit{w}}$, vector of weights of noncentral chi-square components ${\displaystyle \mathit{k}}$, vector of degrees of freedom of noncentral chi-square components ${\displaystyle \mathit{\lambda }}$, vector of non-centrality parameters of chi-square components ${\displaystyle s}$, scale of normal term ${\displaystyle m}$, offset
Support	${\displaystyle x\in \{\begin{array}{l}[m,+\mathrm{\infty })\text{ if }{w}_i\ge 0,s=0,\\ (-\mathrm{\infty },m]\text{ if }{w}_i\le 0,s=0,\\ \mathbb{R}\text{ otherwise.}\end{array}}$
PDF	no closed-form expression
CDF	no closed-form expression
Mean	${\displaystyle \sum _j{w}_j(k_j+{\lambda }_j)+m}$
Variance	${\displaystyle 2\sum _j{w}_j^2(k_j+2{\lambda }_j)+s^2}$
MGF	${\displaystyle \frac{\exp \left[t\left(m+\sum _j\frac{w_j{\lambda }_j}{1-2w_{j}t}\right)+\frac{s^2{t}^2}{2}\right]}{\prod _j{\left(1-2w_{j}t\right)}^{k_j/2}}}$
CF	${\displaystyle \frac{\exp \left[it\left(m+\sum _j\frac{w_j{\lambda }_j}{1-2i{w}_{j}t}\right)-\frac{s^2{t}^2}{2}\right]}{\prod _j{\left(1-2i{w}_{j}t\right)}^{k_j/2}}}$

Inprobability theory andstatistics, thegeneralized chi-squared distribution (orgeneralized chi-square distribution) is the distribution of a quadratic function of amultinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independentnoncentral chi-square variables and anormal variable. There are several other such generalizations for which the same term is sometimes used; some of them are special cases of the family discussed here, for example thegamma distribution.

The generalized chi-squared variable may be described in multiple ways. One is to write it as a weighted sum of independentnoncentral chi-square variables${\displaystyle {{\chi }^{\prime }}^2}$ and a standard normal variable${\displaystyle z}$:^[1][2][3][4]

${\displaystyle \tilde{\chi }(\mathit{w},\mathit{k},\mathit{\lambda },s,m)=\sum _i{w}_i{{\chi }^{\prime }}^2(k_i,{\lambda }_i)+sz+m.}$

Here the parameters are the weights${\displaystyle w_i}$, the degrees of freedom${\displaystyle k_i}$ and non-centralities${\displaystyle {\lambda }_i}$ of the constituent non-central chi-squares, and the coefficients${\displaystyle s}$ and${\displaystyle m}$ of the normal. Some important special cases of this have all weights${\displaystyle w_i}$ of the same sign, or have central chi-squared components, or omit the normal term.

Since a non-central chi-squared variable is a sum of squares of normal variables with different means, the generalized chi-square variable is also defined as a sum of squares of independent normal variables, plus an independent normal variable: that is, a quadratic in normal variables.

Another equivalent way is to formulate it as a quadratic form of a normal vector${\displaystyle \mathit{x}}$:^[5][6][4][3]

${\displaystyle \tilde{\chi }=q(\mathit{x})={\mathit{x}}^{\prime }{\mathbf{Q}}_{\mathbf{2}}\mathit{x}+{{\mathit{q}}_{\mathbf{1}}}^{\prime }\mathit{x}+q_0}$.

Here${\displaystyle {\mathbf{Q}}_{\mathbf{2}}}$ is a matrix,${\displaystyle {\mathit{q}}_{\mathbf{1}}}$ is a vector, and${\displaystyle q_0}$ is a scalar. These, together with the mean${\displaystyle \mathit{\mu }}$ and covariance matrix${\displaystyle \mathbf{\Sigma }}$ of the normal vector${\displaystyle \mathit{x}}$, parameterize the distribution.

For the most general case, a reduction towards a common standard form can be made by using a representation of the following form:^[7]

${\displaystyle X=(z+a{)}^{\mathrm{T}}A(z+a)+c^{\mathrm{T}}z=(x+b{)}^{\mathrm{T}}D(x+b)+d^{\mathrm{T}}x+e,}$

whereD is a diagonal matrix and wherex represents a vector of uncorrelatedstandard normal random variables.

A generalized chi-square variable or distribution can be parameterized in two ways. The first is in terms of the weights${\displaystyle w_i}$, the degrees of freedom${\displaystyle k_i}$ and non-centralities${\displaystyle {\lambda }_i}$ of the constituent non-central chi-squares, and the coefficients${\displaystyle s}$ and${\displaystyle m}$ of the added normal term. The second parameterization is using the quadratic form of a normal vector, where the paremeters are the matrix${\displaystyle {\mathbf{Q}}_{\mathbf{2}}}$, the vector${\displaystyle {\mathit{q}}_{\mathbf{1}}}$, and the scalar${\displaystyle q_0}$, and the mean${\displaystyle \mathit{\mu }}$ and covariance matrix${\displaystyle \mathbf{\Sigma }}$ of the normal vector.

The parameters of the first expression (in terms of non-central chi-squares, a normal and a constant) can be calculated in terms of the parameters of the second expression (quadratic form of a normal vector).^[6]

The parameters of the second expression (quadratic form of a normal vector) can also be calculated in terms of the parameters of the first expression (in terms of non-central chi-squares, a normal and a constant).^[4]

There existsMatlab code to convert from one set of parameters to another.

When${\displaystyle s=0}$ and${\displaystyle w_i}$ are all positive or negative, the quadratic is an ellipse. Then the distribution starts from the point${\displaystyle m}$ at one end, which is called a finite tail. The other end tails off at + or${\displaystyle -\mathrm{\infty }}$ respectively, which is called an infinite tail. When${\displaystyle w_i}$ have mixed signs, and/or there is a normal${\displaystyle s}$ term, both tails are infinite, and the support is the entire real line.The methods to compute the CDF and PDF of the distribution behave differently in finite vs. infinite tails (see table below for best method to use in each case).^[4][3]

The probability density, cumulative distribution, and inverse cumulative distribution functions of a generalized chi-squared variable do not have simple closed-form expressions. But there exist several methods to compute them numerically: Ruben's method,^[8] Imhof's method,^[9] IFFT method,^[4][3] ray method,^[4][3] ellipse approximation,^[4][3] infinite-tail (or simply 'tail') approximation,^[4][3] Pearson and extended Pearson's approximations^[4][3] and Liu-Tang-Zhang approximation.

Numerical algorithms^{[7][2][9][6][4][3]} and computer code (Fortran and C,Matlab,R,Python,Julia) have been published that implement some of these methods to compute the PDF, CDF, and inverse CDF, and to generate random numbers.

The following table shows the best methods to use to compute the CDF and PDF for the different parts of the generalized chi-square distribution in different cases.^[4][3] 'Tail' refers to the infinite-tail approximation.

${\displaystyle \tilde{\chi }}$ type	part	best cdf/pdf method(s)
ellipse: ${\displaystyle w_i}$ same sign, ${\displaystyle s=0}$	body	Ruben, Imhof, IFFT, ray
	finite tail	Ruben, ray (if${\displaystyle {\lambda }_i=0}$), ellipse
	infinite tail	Ruben, ray, tail
not ellipse: ${\displaystyle w_i}$ mixed signs, and/or${\displaystyle s\ne 0}$	body	Imhof, IFFT, ray
	infinite tails	ray, tail
sphere: non-central${\displaystyle {\chi }^2}$ (only one term)	body	Matlab'sncx2cdf/ncx2pdf
	finite tail	ncx2cdf/ncx2pdf, ellipse
	infinite tail	ncx2pdf, ray, tail

Asymptotic expressions for the PDF and CDF of the distribution in the lower or upper infinite tail are given by the infinite-tail approximation:^[4][3]

${\displaystyle \underset{x\to \pm \mathrm{\infty }}{lim}f(x)=\frac{a}{\left|w_{\ast }\right|}{f}_{{{\chi }^{\prime }}_{k_{\ast },{\lambda }_{\ast }}^2}\left(\frac{x}{w_{\ast }}\right),}$

${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}F(x)=\underset{x\to \mathrm{\infty }}{lim}\overline{F}(x)=a{\overline{F}}_{{{\chi }^{\prime }}_{k_{\ast },{\lambda }_{\ast }}^2}\left(\frac{x}{w_{\ast }}\right)=a{Q}_{k_{\ast }/2}(\sqrt{{\lambda }_{\ast }},\sqrt{x/w_{\ast }}),}$

${\displaystyle \text{where the constant }a=e^{\frac{m}{2w_{\ast }}+\frac{s^2}{8w_{\ast }^2}}\prod _{j\ne \ast }\frac{\exp \frac{{\lambda }_j{w}_j}{2(w_{\ast }-w_j)}}{{\left(1-\frac{w_j}{w_{\ast }}\right)}^{k_j/2}}.}$

Here,${\displaystyle w_{\ast }}$ is the largest positive or negative weight if we are looking at the upper or lower tail respectively, and${\displaystyle k_{\ast }}$ and${\displaystyle {\lambda }_{\ast }}$ are its corresponding degree and non-centrality.${\displaystyle f_{{{\chi }^{\prime }}_{k_{\ast },{\lambda }_{\ast }}^2}}$ and${\displaystyle F_{{{\chi }^{\prime }}_{k_{\ast },{\lambda }_{\ast }}^2}}$ are the PDF and CDF of the non-central chi-square distribution with parameters${\displaystyle k_{\ast }}$ and${\displaystyle {\lambda }_{\ast }}$.${\displaystyle Q}$ is the Marcum Q-function.

In the far tails, these expressions can be further simplified to ones that are then identical for the pdf${\displaystyle f(x)}$ and tail CDF${\displaystyle p(x)}$ (which is the CDF at a point in the lower tail, or the complementary CDF at a point in the upper tail):^[4][3]

Here again,${\displaystyle w_{\ast }}$ is the largest positive or negative weight if we are looking at the upper or lower tail respectively.

The generalized chi-squared is the distribution ofstatistical estimates in cases where the usualstatistical theory does not hold, as in the examples below.

If apredictive model is fitted byleast squares, but theresiduals have eitherautocorrelation orheteroscedasticity, then alternative models can be compared (inmodel selection) by relating changes in thesum of squares to anasymptotically valid generalized chi-squared distribution.^[5]

If${\displaystyle \mathit{x}}$ is a normal vector, its log likelihood is aquadratic form of${\displaystyle \mathit{x}}$, and is hence distributed as a generalized chi-squared. The log likelihood ratio that${\displaystyle \mathit{x}}$ arises from one normal distribution versus another is also aquadratic form, so distributed as a generalized chi-squared.^[6]

In Gaussian discriminant analysis, samples from multinormal distributions are optimally separated by using aquadratic classifier, a boundary that is a quadratic function (e.g. the curve defined by setting the likelihood ratio between two Gaussians to 1). The classification error rates of different types (false positives and false negatives) are integrals of the normal distributions within the quadratic regions defined by this classifier. Since this is mathematically equivalent to integrating a quadratic form of a normal vector, the result is an integral of a generalized-chi-squared variable.^[6]

The following application arises in the context ofFourier analysis insignal processing,renewal theory inprobability theory, andmulti-antenna systems inwireless communication. The common factor of these areas is that the sum of exponentially distributed variables is of importance (or identically, the sum of squared magnitudes ofcircularly-symmetric centered complex Gaussian variables).

If${\displaystyle Z_i}$ arekindependent,circularly-symmetric centered complex Gaussian random variables withmean 0 andvariance${\displaystyle {\sigma }_i^2}$, then the random variable

${\displaystyle \tilde{Q}=\sum _{i=1}^k|Z_i{|}^2}$

has a generalized chi-squared distribution of a particular form. The difference from the standard chi-squared distribution is that${\displaystyle Z_i}$ are complex and can have different variances, and the difference from the more general generalized chi-squared distribution is that the relevant scaling matrixA is diagonal. If${\displaystyle \mu ={\sigma }_i^2}$ for alli, then${\displaystyle \tilde{Q}}$, scaled down by${\displaystyle \mu /2}$ (i.e. multiplied by${\displaystyle 2/\mu }$), has achi-squared distribution,${\displaystyle {\chi }^2(2k)}$, also known as anErlang distribution. If${\displaystyle {\sigma }_i^2}$ have distinct values for alli, then${\displaystyle \tilde{Q}}$ has the pdf^[10]

${\displaystyle f(x;k,{\sigma }_1^2,\dots ,{\sigma }_k^2)=\sum _{i=1}^k\frac{e^{-\frac{x}{{\sigma }_i^2}}}{{\sigma }_i^2\prod _{j=1,j\ne i}^k\left(1-\frac{{\sigma }_j^2}{{\sigma }_i^2}\right)}\text{for }x\ge 0.}$

If there are sets of repeated variances among${\displaystyle {\sigma }_i^2}$, assume that they are divided intoM sets, each representing a certain variance value. Denote${\displaystyle \mathbf{r}=(r_1,r_2,\dots ,r_M)}$ to be the number of repetitions in each group. That is, themth set contains${\displaystyle r_m}$ variables that have variance${\displaystyle {\sigma }_m^2.}$ It represents an arbitrary linear combination of independent${\displaystyle {\chi }^2}$-distributed random variables with different degrees of freedom:

${\displaystyle \tilde{Q}=\sum _{m=1}^M{\sigma }_m^2/2\ast Q_m,Q_m\sim {\chi }^2(2r_m).}$

The pdf of${\displaystyle \tilde{Q}}$ is^[11]

${\displaystyle f(x;\mathbf{r},{\sigma }_1^2,\dots {\sigma }_M^2)=\prod _{m=1}^M\frac{1}{{\sigma }_m^{2r_m}}\sum _{k=1}^M\sum _{l=1}^{r_k}\frac{{\mathrm{\Psi }}_{k,l,\mathbf{r}}}{(r_k-l)!}(-x{)}^{r_k-l}{e}^{-\frac{x}{{\sigma }_k^2}},\text{ for }x\ge 0,}$

where

${\displaystyle {\mathrm{\Psi }}_{k,l,\mathbf{r}}=(-1{)}^{r_k-1}\sum _{\mathbf{i}\in {\mathrm{\Omega }}_{k,l}}\prod _{j\ne k}(\genfrac{}{}{0ex}{}{i_j+r_j-1}{i_j}){\left(\frac{1}{{\sigma }_j^2}-\frac{1}{{\sigma }_k^2}\right)}^{-(r_j+i_j)},}$

with${\displaystyle \mathbf{i}=[i_1,\dots ,i_M{]}^T}$ from the set${\displaystyle {\mathrm{\Omega }}_{k,l}}$ ofall partitions of${\displaystyle l-1}$ (with${\displaystyle i_k=0}$) defined as

${\displaystyle {\mathrm{\Omega }}_{k,l}=\left\{[i_1,\dots ,i_m]\in {\mathbb{Z}}^m;\sum _{j=1}^M{i}_j=l-1,i_k=0,i_j\ge 0\text{ for all }j\right\}.}$

• Noncentral chi-squared distribution
• Chi-squared distribution

• Davies, R.B.: Fortran and C source code for "Linear combination of chi-squared random variables"
• Das, A: MATLAB code to compute the statistics, pdf, cdf, inverse cdf and random numbers of the generalized chi-square distribution.

• Continuous distributions

• Articles with short description
• Short description is different from Wikidata