Skewed generalized t distribution


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

where is the beta function, is the location parameter, is the scale parameter, is the skewness parameter, and and are the parameters that control the kurtosis. and 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,
and
These values for and yield a distribution with mean of if and a variance of if. In order for to take on this value however, it must be the case that. Similarly, for to equal the above value,.
The parameterization that yields the simplest functional form of the probability density function sets and. This gives a mean of
and a variance of
The parameter controls the skewness of the distribution. To see this, let denote the mode of the distribution, and
Since, the probability left of the mode, and therefore right of the mode as well, can equal any value in depending on the value of. Thus the skewed generalized t distribution can be highly skewed as well as symmetric. If, then the distribution is negatively skewed. If, then the distribution is positively skewed. If, then the distribution is symmetric.
Finally, and control the kurtosis of the distribution. As and get smaller, the kurtosis increases. Large values of and yield a distribution that is more platykurtic.

Moments

Let be a random variable distributed with the skewed generalized t distribution. The moment, for, is:
The mean, for, is:
The variance, for, is:
The skewness, for, is:
The kurtosis, for, is:

Special Cases

Special and limiting cases of the skewed generalized t distribution include the skewed generalized error distribution, the generalized t distribution introduced by McDonald and Newey, the skewed t proposed by Hansen, the skewed Laplace distribution, the generalized error distribution, a skewed normal distribution, the student t distribution, the skewed Cauchy distribution, the Laplace distribution, the uniform distribution, the normal distribution, and the Cauchy distribution. The graphic below, adapted from Hansen, McDonald, and Newey, shows which parameters should be set to obtain some of the different special values of the skewed generalized t distribution.

Skewed generalized error distribution

The Skewed Generalized Error Distribution has the pdf:
where
gives a mean of. Also
gives a variance of.

Generalized t distribution

The Generalized T Distribution has the pdf:
where
gives a variance of.

Skewed t distribution

The Skewed T Distribution has the pdf:
where
gives a mean of. Also
gives a variance of.

Skewed Laplace distribution

The Skewed Laplace Distribution has the pdf:
where
gives a mean of. Also
gives a variance of.

Generalized error distribution

The Generalized Error Distribution has the pdf:
where
gives a variance of.

Skewed normal distribution

The Skewed Normal Distribution has the pdf:
where
gives a mean of. Also
gives a variance of.

Student's t-distribution

The Student's t-distribution has the pdf:
was substituted.

Skewed Cauchy distribution

The Skewed Cauchy Distribution has the pdf:
and was substituted.
The mean, variance, skewness, and kurtosis of the skewed Cauchy distribution are all undefined.

Laplace distribution

The Laplace distribution has the pdf:
was substituted.

Uniform Distribution

The Uniform distribution has the pdf:
Thus the standard uniform parameterization is obtained if,, and.

Normal distribution

The Normal distribution has the pdf:
where
gives a variance of.

Cauchy Distribution

The Cauchy distribution has the pdf:
was substituted.