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A generalized chi-square variable or distribution can be parameterized in two ways. The first is in terms of the weights w i {\displaystyle w_{i}} , the degrees of freedom k i {\displaystyle k_{i}} and non-centralities λ i {\displaystyle \lambda _{i}} of the constituent non-central chi-squares, and the coefficients s {\displaystyle s} and m ...
In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. [1] [2] [3] It is named after the Polish mathematicians Kazimierz Kuratowski and Czesław Ryll-Nardzewski. [4]
The two generalized normal families described here, like the skew normal family, are parametric families that extends the normal distribution by adding a shape parameter. Due to the central role of the normal distribution in probability and statistics, many distributions can be characterized in terms of their relationship to the normal ...
In probability and statistics, the generalized K-distribution is a three-parameter family of continuous probability distributions. The distribution arises by compounding two gamma distributions . In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are:
In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions.It is often used to model the tails of another distribution. It is specified by three parameters: location , scale , and shape
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding:
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral. [1] The density function of , …, is
In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions . Generalized functions are especially useful for treating discontinuous functions more like smooth functions , and describing discrete physical ...