Generalized chi-squared distribution

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.

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

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