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^[1] in 1998. The distribution has since been used in different applications.^[2]^[3]^[4]^[5]^[6]^[7] There are different parameterizations for the skewed generalized t distribution,^[1]^[5] which we account for in this article.

Definition

Probability density function

$f_{SGT}(x;\mu ,\sigma ,\lambda ,p,q)={\frac {p}{2v\sigma q^{1/p}B({\frac {1}{p}},q)({\frac {|x-\mu +m|^{p}}{q(v\sigma )^{p}(\lambda sign(x-\mu +m)+1)^{p}}}+1)^{{\frac {1}{p}}+q}}}$

where $B$ is the beta function, $\mu$ is the location parameter, $\sigma >0$ is the scale parameter, $-1<\lambda <1$ is the skewness parameter, and $p>0$ and $q>0$ are the parameters that control the kurtosis. Note that $m$ and $v$ 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^[1] of the skewed generalized t distribution,

m={\frac {2v\sigma \lambda q^{\frac {1}{p}}B({\frac {2}{p}},q-{\frac {1}{p}})}{B({\frac {1}{p}},q)}}

and

v={\frac {q^{-{\frac {1}{p}}}}{\sqrt {(3\lambda ^{2}+1){\frac {B({\frac {3}{p}},q-{\frac {2}{p}})}{B({\frac {1}{p}},q)}}-4\lambda ^{2}{\frac {B({\frac {2}{p}},q-{\frac {1}{p}})^{2}}{B({\frac {1}{p}},q)^{2}}}}}}

These values for $m$ and $v$ yield a distribution with mean of $\mu$ if $pq>1$ and a variance of $\sigma ^{2}$ if $pq>2$ . In order for $m$ to take on this value however, it must be the case that $pq>1$ . Similarly, for $v$ to equal the above value, $pq>2$ .

The parameterization that yields the simplest functional form of the probability density function sets $m=0$ and $v=1$ . This gives a mean of

\mu +{\frac {2v\sigma \lambda q^{\frac {1}{p}}B({\frac {2}{p}},q-{\frac {1}{p}})}{B({\frac {1}{p}},q)}}

and a variance of

\sigma ^{2}q^{\frac {2}{p}}((3\lambda ^{2}+1){\frac {B({\frac {3}{p}},q-{\frac {2}{p}})}{B({\frac {1}{p}},q)}}-4\lambda ^{2}{\frac {B({\frac {2}{p}},q-{\frac {1}{p}})^{2}}{B({\frac {1}{p}},q)^{2}}})

The $\lambda$ parameter controls the skewness of the distribution. To see this, let $M$ denote the mode of the distribution, and note that

\int _{-\infty }^{M}f_{SGT}(x;\mu ,\sigma ,\lambda ,p,q)dx={\frac {1-\lambda }{2}}

Since $-1<\lambda <1$ , the probability left of the mode, and therefore right of the mode as well, can equal any value in (0,1) depending on the value of $\lambda$ . Thus the skewed generalized t distribution can be highly skewed as well as symmetric. If $-1<\lambda <0$ , then the distribution is negatively skewed. If $0 < \lambda < 1$ , then the distribution is positively skewed. If $\lambda =0$ , then the distribution is symmetric.

Finally, $p$ and $q$ control the kurtosis of the distribution. As $p$ and $q$ get smaller, the kurtosis increases^[1] (i.e. becomes more leptokurtic). Large values of $p$ and $q$ yield a distribution that is more platykurtic.

Moments

Let $X$ be a random variable distributed with the skewed generalized t distribution.The $h^{th}$ moment (i.e. $E[(X-E(X))^{h}]$ ), for $pq>h$ , is: $\sum _{r=0}^{h}{\binom {h}{r}}((1+\lambda )^{r+1}+(-1)^{r}(1-\lambda )^{r+1})(-\lambda )^{h-r}{\frac {(v\sigma )^{h}q^{\frac {h}{p}}B({\frac {r+1}{p}},q-{\frac {r}{p}})B({\frac {2}{p}},q-{\frac {1}{p}})^{h-r}}{2^{r-h+1}B({\frac {1}{p}},q)^{h-r+1}}}$

The mean, for $pq>1$ , is:

\mu +{\frac {2v\sigma \lambda q^{\frac {1}{p}}B({\frac {2}{p}},q-{\frac {1}{p}})}{B({\frac {1}{p}},q)}}-m

The variance (i.e. $E[(X-E(X))^{2}]$ ), for $pq>2$ , is:

(v\sigma )^{2}q^{\frac {2}{p}}((3\lambda ^{2}+1){\frac {B({\frac {3}{p}},q-{\frac {2}{p}})}{B({\frac {1}{p}},q)}}-4\lambda ^{2}{\frac {B({\frac {2}{p}},q-{\frac {1}{p}})^{2}}{B({\frac {1}{p}},q)^{2}}})

The skewness (i.e. $E[(X-E(X))^{3}]$ ), for $pq>3$ , is:

{\frac {2q^{3/p}\lambda (v\sigma )^{3}}{B({\frac {1}{p}},q)^{3}}}{\Bigg (}8\lambda ^{2}B({\frac {2}{p}},q-{\frac {1}{p}})^{3}-3(1+3\lambda ^{2})B({\frac {1}{p}},q)

\times B({\frac {2}{p}},q-{\frac {1}{p}})B({\frac {3}{p}},q-{\frac {2}{p}})+2(1+\lambda ^{2})B({\frac {1}{p}},q)^{2}B({\frac {4}{p}},q-{\frac {3}{p}}){\Bigg )}

The kurtosis (i.e. $E[(X-E(X))^{4}]$ ), for $pq>4$ , is:

{\frac {q^{4/p}(v\sigma )^{4}}{B({\frac {1}{p}},q)^{4}}}{\Bigg (}-48\lambda ^{4}B({\frac {2}{p}},q-{\frac {1}{p}})^{4}+24\lambda ^{2}(1+3\lambda ^{2})B({\frac {1}{p}},q)B({\frac {2}{p}},q-{\frac {1}{p}})^{2}

\times B({\frac {3}{p}},q-{\frac {2}{p}})-32\lambda ^{2}(1+\lambda ^{2})B({\frac {1}{p}},q)^{2}B({\frac {2}{p}},q-{\frac {1}{p}})B({\frac {4}{p}},q-{\frac {3}{p}})

+(1+10\lambda ^{2}+5\lambda ^{4})B({\frac {1}{p}},q)^{3}B({\frac {5}{p}},q-{\frac {4}{p}}){\Bigg )}

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,^[6] the skewed t proposed by Hansen,^[8] the skewed Laplace distribution, the generalized error distribution (also known as the generalized normal 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,^[2] shows which parameters should be set to obtain some of the different special values of the skewed generalized t distribution.

The skewed generalized t distribution tree

skewed generalized error distribution

The Skewed Generalized Error Distribution has the pdf:

\lim _{q\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda ,p,q)

=f_{SGED}(x;\mu ,\sigma ,\lambda ,p)={\frac {pe^{-({\frac {|x-\mu +m|}{v\sigma (1+\lambda sign(x-\mu +m))}})^{p}}}{2v\sigma \Gamma (1/p)}}

where

m={\frac {2^{\frac {2}{p}}v\sigma \lambda \Gamma ({\frac {1}{2}}+{\frac {1}{p}})}{\sqrt {\pi }}}

gives a mean of $\mu$ . Also

v={\sqrt {\frac {\pi \Gamma ({\frac {1}{p}})}{\pi (1+3\lambda ^{2})\Gamma ({\frac {3}{p}})-16^{\frac {1}{p}}\lambda ^{2}\Gamma ({\frac {1}{2}}+{\frac {1}{p}})^{2}\Gamma ({\frac {1}{p}})}}}

gives a variance of $\sigma ^{2}$ .

generalized t distribution

The Generalized T Distribution has the pdf:

f_{SGT}(x;\mu ,\sigma ,\lambda =0,p,q)

=f_{GT}(x;\mu ,\sigma ,p,q)={\frac {p}{2v\sigma q^{1/p}B({\frac {1}{p}},q)({\frac {\left|x-\mu \right|^{p}}{q(v\sigma )^{p}}}+1)^{{\frac {1}{p}}+q}}}

where

v={\frac {1}{q^{1/p}}}{\sqrt {{\frac {B({\frac {1}{p}},q)}{B({\frac {3}{p}},q-{\frac {2}{p}})}})}}

gives a variance of $\sigma ^{2}$ .

skewed t distribution

The Skewed T Distribution has the pdf:

f_{SGT}(x;\mu ,\sigma ,\lambda ,p=2,q)

=f_{ST}(x;\mu ,\sigma ,\lambda ,q)={\frac {\Gamma ({\frac {1}{2}}+q)}{v\sigma (\pi q)^{1/2}\Gamma (q)({\frac {\left|x-\mu +m\right|^{2}}{q(v\sigma )^{2}(\lambda sign(x-\mu +m)+1)^{2}}}+1)^{{\frac {1}{2}}+q}}}

where

m={\frac {2v\sigma \lambda q^{1/2}\Gamma (q-{\frac {1}{2}})}{\pi ^{1/2}\Gamma (q+{\frac {1}{2}})}}

gives a mean of $\mu$ . Also

v={\frac {1}{q^{1/2}{\sqrt {(3\lambda ^{2}+1)({\frac {1}{2q-2}})-{\frac {4\lambda ^{2}}{\pi }}({\frac {\Gamma (q-{\frac {1}{2}})}{\Gamma (q)}})^{2}}}}}

gives a variance of $\sigma ^{2}$ .

skewed Laplace distribution

The Skewed Laplace Distribution has the pdf:

\lim _{q\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda ,p=1,q)

=f_{SLaplace}(x;\mu ,\sigma ,\lambda )={\frac {e^{\frac {-|x-\mu +m|}{v\sigma (1+\lambda sign(x-\mu +m))}}}{2v\sigma }}

where

m=2v\sigma \lambda

gives a mean of $\mu$ . Also

v=[2(1+\lambda ^{2})]^{-{\frac {1}{2}}}

gives a variance of $\sigma ^{2}$ .

generalized error distribution

The Generalized Error Distribution (also known as the generalized normal distribution) has the pdf:

\lim _{q\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda =0,p,q)

=f_{GED}(x;\mu ,\sigma ,p)={\frac {pe^{-({\frac {|x-\mu |}{v\sigma }})^{p}}}{2v\sigma \Gamma (1/p)}}

where

v={\sqrt {\frac {\Gamma ({\frac {1}{p}})}{\Gamma ({\frac {3}{p}})}}}

gives a variance of $\sigma ^{2}$ .

skewed normal distribution

The Skewed Normal Distribution has the pdf:

\lim _{q\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda ,p=2,q)

=f_{SNormal}(x;\mu ,\sigma ,\lambda )={\frac {e^{-({\frac {|x-\mu +m|}{v\sigma (1+\lambda sign(x-\mu +m))}})^{2}}}{v\sigma {\sqrt {\pi }}}}

where

m={\frac {2v\sigma \lambda }{\sqrt {\pi }}}

gives a mean of $\mu$ . Also

v={\sqrt {\frac {2\pi }{(\pi -8\lambda ^{2}+3\pi \lambda ^{2})}}}

gives a variance of $\sigma ^{2}$ .

student's t-distribution

The Student's t-distribution has the pdf:

f_{SGT}(x;\mu =0,\sigma =1,\lambda =0,p=2,q=d/2)

=f_{T}(x;d)={\frac {\Gamma ({\frac {d+1}{2}})}{(\pi d)^{1/2}\Gamma (d/2)({\frac {x^{2}}{d}}+1)^{\frac {d+1}{2}}}}

Note that we substituted $v={\sqrt {2}}$ .

skewed Cauchy distribution

The Skewed Cauchy Distribution has the pdf:

f_{SGT}(x;\mu ,\sigma ,\lambda ,p=2,q=1/2)

=f_{SCauchy}(x;\mu ,\sigma ,\lambda )={\frac {1}{\sigma \pi ({\frac {\left|x-\mu \right|^{2}}{\sigma ^{2}(\lambda sign(x-\mu )+1)^{2}}}+1)}}

Note that we substituted $v={\sqrt {2}}$ and $m=0$ .

Laplace distribution

The Laplace distribution has the pdf:

\lim _{q\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda =0,p=1,q)

=f_{Laplace}(x;\mu ,\sigma )={\frac {e^{\frac {-|x-\mu |}{\sigma }}}{2\sigma }}

Note that we substituted $v=1$ .

Uniform Distribution

The Uniform distribution has the pdf:

\lim _{p\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda ,p,q)

=f(x)={\begin{cases}{\frac {1}{2v\sigma }}&|x-\mu |<v\sigma \\0&\mathrm {otherwise} \end{cases}}

Thus the standard uniform parameterization is obtained if $\mu ={\frac {a+b}{2}}$ , $v=1$ , and $\sigma ={\frac {b-a}{2}}$ .

Normal distribution

The Normal distribution has the pdf:

\lim _{q\to \infty }f_{SGT}(x;\mu ,\sigma ,\lambda =0,p=2,q)

=f_{Normal}(x;\mu ,\sigma )={\frac {e^{-({\frac {|x-\mu |}{v\sigma }})^{2}}}{v\sigma {\sqrt {\pi }}}}

where

v={\sqrt {2}}

gives a variance of $\sigma ^{2}$ .

Cauchy Distribution

The Cauchy distribution has the pdf:

f_{SGT}(x;\mu ,\sigma ,\lambda =0,p=2,q=1/2)

=f_{Cauchy}(x;\mu ,\sigma )={\frac {1}{\sigma \pi (({\frac {x-\mu }{\sigma }})^{2}+1)}}

Note that we substituted $v={\sqrt {2}}$ .

References

Hansen, B. (1994). "Autoregressive Conditional Density Estimation". International Economic Review. 35: 705–730. doi:10.2307/2527081.
Hansen, C.; McDonald, J.; Newey, W. (2010). "Enstrumental Variables Estimation with Flexible Distributions". Journal of Business and Economic Statistics. 28: 13–25. doi:10.1198/jbes.2009.06161.
Hansen, C.; McDonald, J.; Theodossiou, P. (2007). "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models". economics-ejournal.
McDonald, J.; Michefelder, R.; Theodossiou, P. (2009). "Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application". Multinational Finance Journal. 15: 293–321.
McDonald, J.; Michelfelder, R.; Theodossiou, P. (2010). "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas". Quantitative Finance: 375–387.
McDonald, J.; Newey, W. (1988). "Partially Adaptive Estimation of Regression Models via the Generalized t Distribution". Econometric Theory. 4: 428–457. doi:10.1017/s0266466600013384.
Savva, C.; Theodossiou, P. (2015). "Skewness and the Relation between Risk and Return". Management Science.
Theodossiou, P. (1998). "Financial Data and the Skewed Generalized T Distribution". Management Science. 44: 1650–1661. doi:10.1287/mnsc.44.12.1650.

External links

outlines skewed generalized t distribution, its special cases, and a program to calculate its pdf, cdf, and critical values

Notes

1 2 3 4 Theodossiou, P (1998). "Financial Data and the Skewed Generalized T Distribution". Management Science. 44: 1650–1661. doi:10.1287/mnsc.44.12.1650.
1 2 Hansen, C.; McDonald, J.; Newey, W. (2010). "Enstrumental Variables Estimation with Flexible Distributions". Journal of Business and Economic Statistics. 28: 13–25. doi:10.1198/jbes.2009.06161.
↑ Hansen, C., J. McDonald, and P. Theodossiou (2007) "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models" economics-ejournal
↑ McDonald, J.; Michelfelder, R.; Theodossiou, P. (2009). "Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application". Multinational Finance Journal. 15: 293–321.
1 2 McDonald J., R. Michelfelder, and P. Theodossiou (2010) "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas" Quantitative Finance 375-387.
1 2 McDonald, J.; Newey, W. (1998). "Partially Adaptive Estimation of Regression Models via the Generalized t Distribution". Econometric Theory. 4: 428–457.
↑ Savva C. and P. Theodossiou (2015) "Skewness and the Relation between Risk and Return" Management Science, forthcoming.
↑ Hansen, B (1994). "Autoregressive Conditional Density Estimation". International Economic Review. 35: 705–730. doi:10.2307/2527081.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

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