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Concept
Generalized chi-squared distribution
Summary
In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic form of a multinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-square variables and a normal 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 the gamma distribution.
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:
Here the parameters are the weights , the degrees of freedom and non-centralities of the constituent chi-squares, and the normal parameters and . Some important special cases of this have all weights 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 :
Here is a matrix, is a vector, and is a scalar. These, together with the mean and covariance matrix of the normal vector , parameterize the distribution. The parameters of the former expression (in terms of non-central chi-squares, a normal and a constant) can be calculated in terms of the parameters of the latter expression (quadratic form of a normal vector). If (and only if) in this formulation is positive-definite, then all the in the first 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:
where D is a diagonal matrix and where x represents a vector of uncorrelated standard normal random variables.
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