Skewed generalized t distribution

$f\_\backslash text\{SGT\}(x;\; \backslash mu,\; \backslash sigma,\; \backslash lambda,\; p,\; q)\; =\; \backslash frac\{p\}\{2\; v\; \backslash sigma\; q^\{\backslash frac\{1\}\{p\}\}\; B(\backslash frac\{1\}\{p\},q)\; \backslash left[1\; +\; \backslash frac\{|\; x-\backslash mu\; +\; m\; |^p\}\{q\; (v\; \backslash sigma)^p\; (1\; +\; \backslash lambda\; \backslash sgn(x-\backslash mu\; +\; m))^p\}\backslash right]^\{\backslash frac\{1\}\{p\}+q\}\}$

where $B$ is the beta function, $\backslash mu$ is the location parameter, $\backslash sigma\; >\; 0$ is the scale parameter, is the skewness parameter, and $p\; >\; 0$ and $q\; >\; 0$ are the parameters that control the kurtosis. $m$ 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 of the skewed generalized t distribution,

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

and

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

These values for $m$ and $v$ yield a distribution with mean of $\backslash mu$ if $pq\; >\; 1$ and a variance of $\backslash sigma^2$ 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

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

and a variance of

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

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

:$$

\int_{-\infty}^{M} f_\text{SGT}(x; \mu, \sigma, \lambda, p, q) \mathrm{d}x = \frac{1-\lambda}{2}

Since , 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 $\backslash lambda$. 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 $\backslash lambda\; =\; 0$, then the distribution is symmetric.

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

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{2 v \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}} ( (1 + 3 \lambda^2) \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{2 q^{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 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 (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, shows which parameters should be set to obtain some of the different special values of the skewed generalized t distribution.

File:SGTtree.png

The Skewed Generalized Error Distribution (SGED) has the pdf:

:$$

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

:$$

= f_\text{SGED}(x; \mu, \sigma, \lambda, p) = \frac{p}{2 v \sigma \Gamma ( \frac{1}{p} )} e^{- \left( \frac

where

:$$

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

gives a mean of $\backslash mu$. 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 $\backslash sigma^2$.

The generalized t-distribution (GT) has the pdf:

:$$

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

:$$

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

where

:$$

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

gives a variance of $\backslash sigma^2$.

The skewed t-distribution (ST) has the pdf:

:$$

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

:$$

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

where

:$$

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

gives a mean of $\backslash mu$. Also

:$$

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

gives a variance of $\backslash sigma^2$.

The skewed Laplace distribution (SLaplace) has the pdf:

:$$

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

:$$

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

where

:$$

m = 2 v \sigma \lambda

gives a mean of $\backslash mu$. Also

:$$

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

gives a variance of $\backslash sigma^2$.

The generalized error distribution (GED, also known as the generalized normal distribution) has the pdf:

:$$

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

:$$

= f_\text{GED}(x; \mu, \sigma, p) = \frac{p}{2 v \sigma \Gamma ( \frac{1}{p} )} e^{- \left( \frac

where

:$$

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

gives a variance of $\backslash sigma^2$.

The skewed normal distribution (SNormal) has the pdf:

:$$

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

:$$

= f_\text{SNormal}(x; \mu, \sigma, \lambda) = \frac{1}{v \sigma \sqrt{\pi}} e^{- \left[ \frac

where

:$$

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

gives a mean of $\backslash mu$. Also

:$$

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

gives a variance of $\backslash sigma^2$.

The distribution should not be confused with the skew normal distribution or another asymmetric version. Indeed, the distribution here is a special case of a bi-Gaussian, whose left and right widths are proportional to $1-\backslash lambda$ and $1+\backslash lambda$.

The Student's t-distribution (T) has the pdf:

:$$

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

:$$

= f_\text{T}(x; d) = \frac{\Gamma ( \frac{d+1}{2} )}{ (\pi d)^{\frac{1}{2}} \Gamma (\frac{d}{2})} \left(1 + \frac{x^2}{d}\right)^{-\frac{d+1}{2}}

$v\; =\; \backslash sqrt\{2\}$ was substituted.

The skewed cauchy distribution (SCauchy) has the pdf:

:$$

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

:$$

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

$v\; =\; \backslash sqrt\{2\}$ and $m\; =\; 0$ was substituted.

The mean, variance, skewness, and kurtosis of the skewed Cauchy distribution are all undefined.

The Laplace distribution has the pdf:

:$$

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

:$$

= f_\text{Laplace}(x; \mu, \sigma) = \frac{1}{2 \sigma } e^{ -\frac

$v\; =\; 1$ was substituted.

The uniform distribution has the pdf:

:$$

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

:$$

= f(x)=\begin{cases}

\frac{1}{2 v \sigma} & |x - \mu| < v \sigma \\

0 & \mathrm{otherwise}

\end{cases}

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

The normal distribution has the pdf:

:$$

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

:$$

= f_\text{Normal}(x; \mu, \sigma) = \frac{e^{- \left( \frac

where

:$$

v = \sqrt{2}

gives a variance of $\backslash sigma^2$.

The Cauchy distribution has the pdf:

:$$

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

:$$

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

$v\; =\; \backslash sqrt\{2\}$ was substituted.

• {{Cite journal

| last1 = Hansen | first1 = B.

| year = 1994

| title = Autoregressive Conditional Density Estimation

| journal = International Economic Review

| volume = 35

| issue = 3

| pages = 705–730

| doi=10.2307/2527081

| jstor = 2527081

}}

• {{Cite journal

| last1 = Hansen | first1 = C.

| last2 = McDonald | first2 = J.

| last3 = Newey | first3 = W.

| year = 2010

| title = Instrumental Variables Estimation with Flexible Distributions

| journal = Journal of Business and Economic Statistics

| volume = 28

| pages = 13–25

| doi=10.1198/jbes.2009.06161

| hdl = 10419/79273

| s2cid = 11370711

| hdl-access = free}}

• {{Cite journal

| last1 = Hansen | first1 = C.

| last2 = McDonald | first2 = J.

| last3 = Theodossiou | first3 = P.

| year = 2007

| title = Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models

| journal = Economics: The Open-Access, Open-Assessment e-Journal

| volume = 1

| issue = 2007–7

| pages = 1

| doi = 10.5018/economics-ejournal.ja.2007-7

| doi-access = free

| hdl = 20.500.14279/1024

| hdl-access = free

}}

• {{Cite journal

| last1 = McDonald | first1 = J.

| last2 = Michefelder | first2 = R.

| last3 = Theodossiou | first3 = P.

| year = 2009

| title = Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application

| journal = Multinational Finance Journal

| volume = 15

| issue = 3/4

| pages = 293–321

| doi = 10.17578/13-3/4-6

| s2cid = 15012865

| url = http://www.mfsociety.org/modules/modDashboard/uploadFiles/journals/MJ~777~p16uelq1mr19251cptdtu1nab1lsr4.pdf

}}

• {{Cite journal

| last1 = McDonald | first1 = J.

| last2 = Michelfelder | first2 = R.

| last3 = Theodossiou | first3 = P.

| year = 2010

| title = Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas

| journal = Quantitative Finance

| volume = 10

| issue = 4

| pages = 375–387

| doi = 10.1080/14697680902814241

| s2cid = 11130911

}}

• {{Cite journal

| last1 = McDonald | first1 = J.

| last2 = Newey | first2 = W.

| year = 1988

| title = Partially Adaptive Estimation of Regression Models via the Generalized t Distribution

| journal = Econometric Theory

| volume = 4

| issue = 3

| pages = 428–457

| doi=10.1017/s0266466600013384

| s2cid = 120305707

}}

• {{Cite journal

| last1 = Savva | first1 = C.

| last2 = Theodossiou | first2 = P.

| year = 2015

| title = Skewness and the Relation between Risk and Return

|journal = Management Science

}}

• {{Cite journal

| last1 = Theodossiou | first1 = P.

| year = 1998

| title = Financial Data and the Skewed Generalized T Distribution

|journal = Management Science

| volume = 44

| issue = 12–part–1

| pages = 1650–1661

| doi=10.1287/mnsc.44.12.1650

}}

• [https://cran.r-project.org/web/packages/sgt/vignettes/sgt.pdf outlines skewed generalized t distribution, its special cases, and a program to calculate its pdf, cdf, and critical values]
• online demo https://www.desmos.com/calculator/hfb11etkeq

{{Reflist|30em|refs=

• {{cite journal | last1 = Hansen | first1 = B | year = 1994 | title = Autoregressive Conditional Density Estimation | journal = International Economic Review | volume = 35 | issue = 3| pages = 705–730 | doi=10.2307/2527081| jstor = 2527081 }}
• {{cite journal | last1 = Hansen | first1 = C. | last2 = McDonald | first2 = J. | last3 = Newey | first3 = W. | year = 2010 | title = Instrumental Variables Estimation with Flexible Distributions | journal = Journal of Business and Economic Statistics | volume = 28 | pages = 13–25 | doi=10.1198/jbes.2009.06161| hdl = 10419/79273 | s2cid = 11370711 | hdl-access = free }}
• Hansen, C., J. McDonald, and P. Theodossiou (2007) "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models" Economics: The Open-Access, Open-Assessment E-Journal
• {{cite journal | last1 = McDonald | first1 = J. | last2 = Michelfelder | first2 = R. | last3 = Theodossiou | first3 = P. | year = 2009 | title = Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application | url =http://www.mfsociety.org/modules/modDashboard/uploadFiles/journals/MJ~777~p16uelq1mr19251cptdtu1nab1lsr4.pdf | journal = Multinational Finance Journal | volume = 15 | issue = 3/4| pages = 293–321 | doi = 10.17578/13-3/4-6 | s2cid = 15012865 }}
• McDonald J., R. Michelfelder, and P. Theodossiou (2010) "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas" Quantitative Finance 375-387.
• {{cite journal | last1 = McDonald | first1 = J. | last2 = Newey | first2 = W. | year = 1998 | title = Partially Adaptive Estimation of Regression Models via the Generalized t Distribution | journal = Econometric Theory | volume = 4 | issue = 3| pages = 428–457 | doi = 10.1017/S0266466600013384 | s2cid = 120305707 }}
• Savva C. and P. Theodossiou (2015) "Skewness and the Relation between Risk and Return" Management Science, forthcoming.
• {{cite journal | last1 = Theodossiou | first1 = P | year = 1998 | title = Financial Data and the Skewed Generalized T Distribution | journal = Management Science | volume = 44 | issue = 12–part–1| pages = 1650–1661 | doi=10.1287/mnsc.44.12.1650}}

}}

{{ProbDistributions|continuous-infinite}}

{{Statistics|state=collapsed}}

{{DEFAULTSORT:Skewed generalized t distribution}}

Category:Continuous distributions

Category:Normal distribution

Category:Probability distributions with non-finite variance

Category:Location-scale family probability distributions