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| <!-- This article was started in British English and so makes extensive use of chi-squared rather than chi-sqaure -->
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| In [[probability theory]] and [[statistics]], the specific name '''generalized chi-squared distribution''' (also '''generalized chi-square distribution''') arises in relation to one particular family of variants of the [[chi-squared distribution]]. There are several other such variants for which the same term is sometimes used, or which clearly are generalizations of the chi-squared distribution, and which are treated elsewhere: some are special cases of the family discussed here, for example the [[noncentral chi-squared distribution]] and the [[gamma distribution]], while the [[generalized gamma distribution]] is outside this family. The type of generalisation of the chi-squared distribution that is discussed here is of importance because it arises in the context of the distribution of [[estimation|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 a statistical analysis of alternative model structures can be undertaken by relating changes in the [[sum of squares]]{{dn|date=December 2013}} to an [[asymptotic distribution|asymptotically valid]] generalized chi-squared distribution.<ref name=Jones1>Jones, D.A. (1983) "Statistical analysis of empirical models fitted by optimisation", [[Biometrika]], 70 (1), 67–88</ref> More specifically, the distribution can be defined in terms of a [[quadratic form (statistics)|quadratic form]] derived from a [[multivariate normal distribution]].
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| ==Definition==
| | These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. There are number of web services that offer Word press development across the world. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Are you considering getting your website redesigned. Our skilled expertise, skillfulness and excellence have been well known all across the world. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>It is the convenient service through which professionals either improve the position or keep the ranking intact. * Robust CRM to control and connect with your subscribers. When we talk about functional suitability, Word - Press proves itself as one of the strongest contestant among its other rivals. If you are looking for Hire Wordpress Developer then just get in touch with him. It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Automated deal feed integration option to populate your blog with relevant deals. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. If this is not possible you still have the choice of the default theme that is Word - Press 3. Get started today so that people searching for your type of business will be directed to you. |
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| One formulation of the generalized chi-squared distribution is as follows.<ref name=Jones1 /> Let ''z'' have a multivariate normal distribution with zero mean and covariance matrix ''B'', then the value of the [[quadratic form (statistics)|quadratic form]] ''X''=''z''<sup>T</sup>Az, where ''A'' is a matrix, has a generalised chi-squared distribution with parameters ''A'' and ''B''. Note that there is some redundancy in this formulation, as for any matrix ''C'', the distribution with parameters ''C''<sup>T</sup>''AC'' and ''B'' is identical to the distribution with parameters ''A'' and ''CBC''<sup>T</sup>. The most general form of generalized chi-squared distribution is obtained by extending the above consideration in two ways: firstly, to allow ''z'' to have a non-zero mean and, secondly, to include an additional linear combination of ''z'' in the definition of ''X''.
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| Note that, in the above formulation, ''A'' and ''B'' need not be [[positive-definite matrix|positive definite]]. However, the case where A is restricted to be at least positive semidefinite is an important one.
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| For the most general case, a reduction towards a common standard form can be made by using a representation of the following form:<ref name=Sheil>Sheil, J., O'Muircheartaigh, I. (1977) "Algorithm AS106: The distribution of non-negative quadratic forms in normal variables",''Applied Statistics'', 26, 92–98</ref>
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| :<math>X=(z+a)^\mathrm T A(z+a)+c^\mathrm T z= (x+b)^\mathrm T D(x+b)+d^\mathrm T x+e ,</math>
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| where ''D'' is a diagonal matrix and where ''x'' represents a vector of uncorrelated [[standard normal]] random variables. An alternative representation can be stated in the form:<ref name=Davies1>Davies, R.B. (1973) Numerical inversion of a characteristic function. [[Biometrika]], 60 (2), 415–417</ref><ref name=Davies2>Davies, R,B. (1980) "Algorithm AS155: The distribution of a linear combination of ''χ''<sup>2</sup> random variables", ''Applied Statistics'', 29, 323–333</ref>
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| :<math>X=\sum_{i=1}^r \lambda_i Y_i + f Z_0 ,</math>
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| where the ''Y<sub>i</sub>'' represent random variables having (different) [[noncentral chi-squared distribution]]s, where ''Z''<sub>0</sub> has a [[standard normal]] distribution, and where all these random variables are independent. Some important special cases relating to this particular form either omit the additional standard normal term and/or have central rather than non-central [[chi-squared distribution]]s for the components of the summation.
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| ==Practical implementation==
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| Computer code for evaluating the [[cumulative distribution function]] of the generalized chi-squared distribution has been published,<ref name=Sheil /><ref name=Davies2 /> but some preliminary manipulation of the parameters of the distribution is usually necessary.
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| ==Other applications==
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| The following application arises in the context of [[Fourier analysis]] in [[signal processing]], [[Renewal theory]] in [[Probability theory]], and [[MIMO|multi-antenna systems]] in [[Wireless|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 [[complex normal distribution|circular symmetric complex Gaussian]] variables).
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| If <math>Z_i</math> are ''k'' [[statistical independence|independent]], [[complex normal distribution|circular symmetric complex Gaussian]] random variables with [[mean]] 0 and [[variance]] <math>\sigma_i^2</math>, then the random variable
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| :<math>\tilde{Q} = \sum_{i=1}^k |Z_i|^2</math>
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| has a generalized chi-squared distribution of a particular form. The difference from the standard chi-squared distribution is that <math>Z_i</math> 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 <math>\mu=\sigma_i^2</math> for all ''i'', then <math>\tilde{Q}</math>, scaled down by <math>\mu/2</math> (i.e. multiplied by <math>2/\mu</math>), has a [[chi-squared distribution]], <math>\chi^2(2k)</math>, also known as an [[Erlang distribution]]. If <math>\sigma_i^2</math> have distinct values for all ''i'', then <math>\tilde{Q}</math> has the pdf<ref>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</ref>
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| :<math>
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| 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
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| i}^{k} (1- \frac{\sigma_j^2}{\sigma_i^2})} \quad\mbox{for }x\geq0.
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| </math>
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| If there are sets of repeated variances among <math>\sigma_i^2</math>, assume that they are divided into ''M'' sets, each representing a certain variance value. Denote <math>\mathbf{r}=(r_1, r_2, \dots, r_M)</math> to be the number of repetitions in each group. That is, the ''m''th set contains <math>r_m</math> variables that have variance <math>\sigma^2_m.</math> It represents an arbitrary linear combination of independent <math>\chi^2</math>-distributed random variables with different degrees of freedom:
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| :<math>\tilde{Q} = \sum_{m=1}^M \sigma^2_m Q_m, \quad Q_m \sim \chi^2(2r_m) \, .</math>
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| The pdf of <math>\tilde{Q}</math> is<ref>E. Björnson, D. Hammarwall, B. Ottersten (2009) [http://kth.diva-portal.org/smash/get/diva2:402940/FULLTEXT01 "Exploiting Quantized Channel Norm Feedback through Conditional Statistics in Arbitrarily Correlated MIMO Systems"], ''IEEE Transactions on Signal Processing'', 57, 4027-4041</ref>
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| :<math>
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| f(x; \mathbf{r}, \sigma^2_1, \dots \sigma^2_M) = \prod_{m=1}^M \frac{1}{\sigma^{2r_m}_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^2_k}} \quad\text{, for }x\geq0 ,
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| </math>
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| where
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| :<math>\Psi_{k,l,\mathbf{r}} = (-1)^{r_k-1} \sum_{\mathbf{i} \in
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| \Omega_{k,l}} \prod_{j \neq k} \Big( \!\!\!
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| \begin{array}{c}
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| i_j + r_j-1\\
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| i_j
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| \end{array} \!\!\! \Big) \Big(\frac{1}{\sigma^2_j}\!-\!\frac{1}{\sigma^2_k} \Big)^{-(r_j + i_j)},
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| </math>
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| with <math>\mathbf{i}=[i_1,\ldots,i_M]^T</math> from the set <math>\Omega_{k,l}</math> of
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| all partitions of <math>l-1</math> (with <math>i_k=0</math>) defined as
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| :<math>
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| \Omega_{k,l} = \Big\{ [i_1,\ldots,i_m]\in \mathbb{Z}^m;
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| \sum_{j=1}^M i_j \!= l-1, i_k=0, i_j\geq 0 \,\, \text{, for all } j
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| \Big\}.</math>
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| ==See also==
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| * [[Degrees of freedom (statistics)#Alternative]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://www.statsresearch.co.nz/robert/QF.htm Davies, R.B.: Fortran and C source code for "Linear combination of chi-squared random variables"]
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| {{ProbDistributions}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. There are number of web services that offer Word press development across the world. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Are you considering getting your website redesigned. Our skilled expertise, skillfulness and excellence have been well known all across the world. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.
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