The skewed generalized 
distribution is a family of continuous statistical
distributions with independently adjustable tail weight, peakedness, and asymmetry.
One convenient nonstandardized parameterization begins with the symmetric generalized
density
![g_(p,q)(y)=p/(2q^(1/p)B(1/p,q))[1+(|y|^p)/q]^(-(q+1/p)),](/images/equations/SkewedGeneralizedtDistribution/NumberedEquation1.svg) |
(1)
|
where 
and 
is the beta function. For location 
, scale 
, and skewness parameter 
, put 
and define
 |
(2)
|
The unequal left and right scales preserve total probability because the two halves have masses 
and 
.
When 
,
the distribution reduces to a symmetric generalized 
distribution. After reparameterization or limiting choices
of 
and 
,
the family includes the Student's t-distribution,
normal distribution, Cauchy
distribution, and Laplace distribution.
Theodossiou (1998) introduced a standardized parameterization for modeling skewness
and heavy tails in financial data.
