In probability and statistics, the skewed generalized "t" distribution is a family of continuous probability distributions. The distribution was first introduced by Panayiotis Theodossiou in 1998. The distribution has since been used in different applications. There are different parameterizations for the skewed generalized t distribution.
Definition
Probability density function
f SGT ( x ; μ , σ , λ , p , q ) = p 2 v σ q 1 p B ( 1 p , q ) [ 1 + | x − μ + m | p q ( v σ ) p ( 1 + λ sgn ( x − μ + m ) ) p ] 1 p + q {\displaystyle f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p,q)={\frac {p}{2v\sigma q^{\frac {1}{p}}B({\frac {1}{p}},q)\left[1+{\frac {|x-\mu +m|^{p}}{q(v\sigma )^{p}(1+\lambda \operatorname {sgn}(x-\mu +m))^{p}}}\right]^{{\frac {1}{p}}+q}}}}
where B {\displaystyle B} is the beta function, μ {\displaystyle \mu } is the location parameter, σ > 0 {\displaystyle \sigma >0} is the scale parameter, − 1 < λ < 1 {\displaystyle -1<\lambda <1} is the skewness parameter, and p > 0 {\displaystyle p>0} and q > 0 {\displaystyle q>0} are the parameters that control the kurtosis. m {\displaystyle m} and v {\displaystyle v} are not parameters, but functions of the other parameters that are used here to scale or shift the distribution appropriately to match the various parameterizations of this distribution. In the original parameterization of the skewed generalized t distribution,
m = λ v σ 2 q 1 p B ( 2 p , q − 1 p ) B ( 1 p , q ) {\displaystyle m=\lambda v\sigma {\frac {2q^{\frac {1}{p}}B({\frac {2}{p}},q-{\frac {1}{p}})}{B({\frac {1}{p}},q)}}}
and
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