Generalized chi-squared distribution

In probability theory and statistics, the generalized chi-squared distribution (also generalized chi-square distribution) is the distribution of a linear sum of independent non-central chi-square variables and a normal variable, or equivalently, of a quadratic form of a multivariate normal distribution. It is a generalization of the noncentral chi-squared distribution. 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 the gamma distribution.

Generalized chi-squared distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle {\boldsymbol {\lambda }}}$, vector of weights of chi-square components ${\displaystyle {\boldsymbol {m}}}$, vector of degrees of freedom of chi-square components ${\displaystyle {\boldsymbol {\delta }}}$, vector of non-centrality parameters of chi-square components ${\displaystyle \sigma }$, scale of normal term
Support	${\displaystyle x\in \mathbb {R} }$
Mean	${\displaystyle \sum \lambda _{j}(m_{j}+\delta _{j}^{2})}$
Variance	${\displaystyle 2\sum \lambda _{j}^{2}(m_{j}+2\delta _{j}^{2})+\sigma ^{2}}$
CF	${\displaystyle {\frac {\exp \left(it\sum {\frac {\lambda _{j}\delta _{j}^{2}}{1-2i\lambda _{j}t}}-{\frac {\sigma ^{2}t^{2}}{2}}\right)}{\prod \left(1-2i\lambda _{j}t\right)^{m_{j}/2}}}}$

Definition

The generalized chi-squared variable may be described in multiple ways. One is to write it as a linear sum of independent noncentral chi-square variables and a normal variable [1][2]:

${\displaystyle \xi =\sum _{i}\lambda _{i}y_{i}+\sigma z,\quad y_{i}\sim \chi '^{2}(m_{i},\delta _{i}^{2}),\quad z\sim N(0,1).}$

Here the parameters are the weights ${\displaystyle \lambda _{i}}$ and ${\displaystyle \sigma }$, and the degrees of freedom ${\displaystyle m_{i}}$ and non-centralities ${\displaystyle \delta _{i}^{2}}$ of the constituent chi-squares. Some important special cases of this have all coefficients the same sign, omit the normal term or have central chi-squared components.

Another is to formulate it as a quadratic form of a normal vector ${\displaystyle {\boldsymbol {x}}}$ [3]:

${\displaystyle \xi =q({\boldsymbol {x}})={\boldsymbol {x}}'\mathbf {Q_{2}} {\boldsymbol {x}}+{\boldsymbol {q_{1}}}'{\boldsymbol {x}}+q_{0}}$.

Here ${\displaystyle \mathbf {Q_{2}} }$ is a matrix, ${\displaystyle {\boldsymbol {q_{1}}}}$ is a vector, and ${\displaystyle q_{0}}$ is a scalar. These, together with the mean ${\displaystyle {\boldsymbol {\mu }}}$ and covariance matrix ${\displaystyle \mathbf {\Sigma } }$ of the normal vector ${\displaystyle {\boldsymbol {x}}}$, parameterize the distribution. If (and only if) ${\displaystyle \mathbf {Q_{2}} }$ in this formulation is positive-definite, all the ${\displaystyle \lambda _{i}}$ in the other formulation will have the same sign.

For the most general case, a reduction towards a common standard form can be made by using a representation of the following form:[4]

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

where D is a diagonal matrix and where x represents a vector of uncorrelated standard normal random variables.

Probability density and cumulative distribution functions

The probability density and cumulative distribution functions of a generalized chi-squared variable do not have simple closed-form expressions. However, numerical algorithms [4][2][5] and computer code (Fortran and C, Matlab, R) for evaluating them have been published.

Applications

The generalized chi-squared is the distribution of statistical estimates in cases where the usual statistical theory does not hold. For example, if a predictive model is fitted by least squares, but the model errors have either autocorrelation or heteroscedasticity, then alternative models can be compared by relating changes in the sum of squares to an asymptotically valid generalized chi-squared distribution.[3]

Classifying normal samples using Gaussian discriminant analysis

If ${\displaystyle {\boldsymbol {x}}}$ is a normal variable, its log likelihood is a quadratic form of ${\displaystyle {\boldsymbol {x}}}$, and is hence distributed as a generalized chi-squared. The log likelihood ratio that ${\displaystyle {\boldsymbol {x}}}$ arises from one normal distribution versus another is also a quadratic form, so distributed as a generalized chi-squared.

In Gaussian discriminant analysis, samples from normal distributions are optimally separated by using a quadratic 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 variable, the result is an integral of a generalized-chi-squared variable.

In signal processing

The following application arises in the context of Fourier analysis in signal processing, renewal theory in probability theory, and multi-antenna systems in wireless 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 circular symmetric complex Gaussian variables).

If ${\displaystyle Z_{i}}$ are k independent, circular symmetric complex Gaussian random variables with mean 0 and variance ${\displaystyle \sigma _{i}^{2}}$, 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 matrix A is diagonal. If ${\displaystyle \mu =\sigma _{i}^{2}}$ for all i, then ${\displaystyle {\tilde {Q}}}$, scaled down by ${\displaystyle \mu /2}$ (i.e. multiplied by ${\displaystyle 2/\mu }$), has a chi-squared distribution, ${\displaystyle \chi ^{2}(2k)}$, also known as an Erlang distribution. If ${\displaystyle \sigma _{i}^{2}}$ have distinct values for all i, then ${\displaystyle {\tilde {Q}}}$ has the pdf[6]

${\displaystyle f(x;k,\sigma _{1}^{2},\ldots ,\sigma _{k}^{2})=\sum _{i=1}^{k}{\frac {e^{-{\frac {x}{\sigma _{i}^{2}}}}}{\sigma _{i}^{2}\prod _{j=1,j\neq i}^{k}\left(1-{\frac {\sigma _{j}^{2}}{\sigma _{i}^{2}}}\right)}}\quad {\text{for }}x\geq 0.}$

If there are sets of repeated variances among ${\displaystyle \sigma _{i}^{2}}$, assume that they are divided into M 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, the mth 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:

${\displaystyle {\tilde {Q}}=\sum _{m=1}^{M}\sigma _{m}^{2}/2*Q_{m},\quad Q_{m}\sim \chi ^{2}(2r_{m})\,.}$

The pdf of ${\displaystyle {\tilde {Q}}}$ is[7]

${\displaystyle f(x;\mathbf {r} ,\sigma _{1}^{2},\dots \sigma _{M}^{2})=\prod _{m=1}^{M}{\frac {1}{\sigma _{m}^{2r_{m}}}}\sum _{k=1}^{M}\sum _{l=1}^{r_{k}}{\frac {\Psi _{k,l,\mathbf {r} }}{(r_{k}-l)!}}(-x)^{r_{k}-l}e^{-{\frac {x}{\sigma _{k}^{2}}}},\quad {\text{ for }}x\geq 0,}$

where

${\displaystyle \Psi _{k,l,\mathbf {r} }=(-1)^{r_{k}-1}\sum _{\mathbf {i} \in \Omega _{k,l}}\prod _{j\neq k}{\binom {i_{j}+r_{j}-1}{i_{j}}}\left({\frac {1}{\sigma _{j}^{2}}}\!-\!{\frac {1}{\sigma _{k}^{2}}}\right)^{-(r_{j}+i_{j})},}$

with ${\displaystyle \mathbf {i} =[i_{1},\ldots ,i_{M}]^{T}}$ from the set ${\displaystyle \Omega _{k,l}}$ of all partitions of ${\displaystyle l-1}$ (with ${\displaystyle i_{k}=0}$) defined as

${\displaystyle \Omega _{k,l}=\left\{[i_{1},\ldots ,i_{m}]\in \mathbb {Z} ^{m};\sum _{j=1}^{M}i_{j}\!=l-1,i_{k}=0,i_{j}\geq 0{\text{ for all }}j\right\}.}$

See also

• Degrees of freedom (statistics)#Alternative
• Noncentral chi-squared distribution
• Chi-squared distribution

References

1. Davies, R.B. (1973) Numerical inversion of a characteristic function. Biometrika, 60 (2), 415–417
2. Davies, R,B. (1980) "Algorithm AS155: The distribution of a linear combination of χ² random variables", Applied Statistics, 29, 323–333
3. Jones, D.A. (1983) "Statistical analysis of empirical models fitted by optimisation", Biometrika, 70 (1), 67–88
4. Sheil, J., O'Muircheartaigh, I. (1977) "Algorithm AS106: The distribution of non-negative quadratic forms in normal variables",Applied Statistics, 26, 92–98
5. Imhof, J. P. (1961). "Computing the Distribution of Quadratic Forms in Normal Variables". Biometrika. 48 (3/4): 419–426. doi:10.2307/2332763. JSTOR 2332763.
6. D. Hammarwall, M. Bengtsson, B. Ottersten (2008) "Acquiring Partial CSI for Spatially Selective Transmission by Instantaneous Channel Norm Feedback", IEEE Transactions on Signal Processing, 56, 1188–1204
7. E. Björnson, D. Hammarwall, B. Ottersten (2009) "Exploiting Quantized Channel Norm Feedback through Conditional Statistics in Arbitrarily Correlated MIMO Systems", IEEE Transactions on Signal Processing, 57, 4027–4041

External links

• Davies, R.B.: Fortran and C source code for "Linear combination of chi-squared random variables"
• Das, A: MATLAB code to compute PDF and CDF of a generalized chi-squared distribution.