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.
fSGT(x;\mu,\sigma,λ,p,q)=
p | |||||||||||||||||||||||
|
where
B
\mu
\sigma>0
-1<λ<1
p>0
q>0
m
v
In the original parameterization of the skewed generalized t distribution,
m=λv\sigma
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|
v=
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m
v
\mu
pq>1
\sigma2
pq>2
m
pq>1
v
pq>2
The parameterization that yields the simplest functional form of the probability density function sets
m=0
v=1
\mu+
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|
\sigma2
| ||||
q |
((1+3λ2)
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|
-4λ2
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)
The
λ
M
M | |
\int | |
-infty |
fSGT(x;\mu,\sigma,λ,p,q)dx=
1-λ | |
2 |
Since
-1<λ<1
λ
-1<λ<0
0<λ<1
λ=0
Finally,
p
q
p
q
p
q
Let
X
hth
E[(X-E(X))h]
pq>h
h | |
\sum | |
r=0 |
\binom{h}{r}((1+λ)r+1+(-1)r(1-λ)r+1)(-λ)h-r
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The mean, for
pq>1
\mu+
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|
-m
The variance (i.e.
E[(X-E(X))2]
pq>2
(v\sigma)2
| ||||
q |
((1+3λ2)
| |||||||||
|
-4λ2
| |||||||||
|
)
The skewness (i.e.
E[(X-E(X))3]
pq>3
2q3/pλ(v\sigma)3 | ||||
|
(8λ2B(
2 | ,q- | |
p |
1 | |
p |
)3-3(1+3λ2)B(
1 | |
p |
,q)
x B(
2 | ,q- | |
p |
1 | |
p |
)B(
3 | ,q- | |
p |
2 | |
p |
)+2 (1+λ2)B(
1 | |
p |
,q)2B(
4 | ,q- | |
p |
3 | |
p |
))
The kurtosis (i.e.
E[(X-E(X))4]
pq>4
q4/p(v\sigma)4 | ||||
|
(-48λ4B(
2 | ,q- | |
p |
1 | |
p |
)4+24 λ2(1+3λ2)B(
1 | |
p |
,q)B(
2 | ,q- | |
p |
1 | |
p |
)2
x B(
3 | ,q- | |
p |
2 | |
p |
)-32 λ2(1+λ2)B(
1 | |
p |
,q)2B(
2 | ,q- | |
p |
1 | |
p |
)B(
4 | ,q- | |
p |
3 | |
p |
)
+(1+10 λ2+5λ4)B(
1 | |
p |
,q)3B(
5 | ,q- | |
p |
4 | |
p |
))
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.
The Skewed Generalized Error Distribution (SGED) has the pdf:
\limq\toinftyfSGT(x;\mu,\sigma,λ,p,q)
=fSGED(x;\mu,\sigma,λ,p)=
p | |||||
|
| ||||||
e |
m=λv\sigma
| ||||||||||||||||||
\sqrt{\pi |
\mu
v=\sqrt{
| |||||||||||||||||||||
|
\sigma2
The generalized t-distribution (GT) has the pdf:
fSGT(x;\mu,\sigma,λ{=}0,p,q)
=fGT(x;\mu,\sigma,p,q)=
p | |||||||||||||||||||||||
|
v=
1 | \sqrt{ | ||||||
|
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|
\sigma2
The skewed t-distribution (ST) has the pdf:
fSGT(x;\mu,\sigma,λ,p{=}2,q)
=fST(x;\mu,\sigma,λ,q)=
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|
m=λv\sigma
| |||||||||||||
|
\mu
v=
1 | |||||||||||||||||||||||
|
\sigma2
The skewed Laplace distribution (SLaplace) has the pdf:
\limq\toinftyfSGT(x;\mu,\sigma,λ,p{=}1,q)
=fSLaplace(x;\mu,\sigma,λ)=
1 | |
2v\sigma |
| ||||||
e |
m=2v\sigmaλ
\mu
v=[2(1+λ2)
| ||||
] |
\sigma2
The generalized error distribution (GED, also known as the generalized normal distribution) has the pdf:
\limq\toinftyfSGT(x;\mu,\sigma,λ{=}0,p,q)
=fGED(x;\mu,\sigma,p)=
p | |||||
|
| ||||||
e |
v=\sqrt{
| ||||||
|
\sigma2
The skewed normal distribution (SNormal) has the pdf:
\limq\toinftyfSGT(x;\mu,\sigma,λ,p{=}2,q)
=fSNormal(x;\mu,\sigma,λ)=
1 | |
v\sigma\sqrt{\pi |
m=λv\sigma
2 | |
\sqrt{\pi |
\mu
v=\sqrt{
2\pi | |
\pi-8λ2+3\piλ2 |
}
\sigma2
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-λ
1+λ
The Student's t-distribution (T) has the pdf:
fSGT(x;\mu{=}0,\sigma{=}1,λ{=}0,p{=}2,q{=}\tfrac{d}{2})
=fT(x;d)=
| ||||||||||||
|
\left(1+
x2 | |
d |
| ||||
\right) |
v=\sqrt{2}
The skewed cauchy distribution (SCauchy) has the pdf:
fSGT(x;\mu,\sigma,λ,p{=}2,q{=}\tfrac{1}{2})
=fSCauchy(x;\mu,\sigma,λ)=
1 | |||||
|
v=\sqrt{2}
m=0
The mean, variance, skewness, and kurtosis of the skewed Cauchy distribution are all undefined.
The Laplace distribution has the pdf:
\limq\toinftyfSGT(x;\mu,\sigma,λ{=}0,p{=}1,q)
=fLaplace(x;\mu,\sigma)=
1 | |
2\sigma |
| ||||||
e |
v=1
The uniform distribution has the pdf:
\limp\toinftyfSGT(x;\mu,\sigma,λ,p,q)
=f(x)=\begin{cases}
1 | |
2v\sigma |
&|x-\mu|<v\sigma\\ 0&otherwise \end{cases}
\mu=
a+b | |
2 |
v=1
\sigma=
b-a | |
2 |
The normal distribution has the pdf:
\limq\toinftyfSGT(x;\mu,\sigma,λ{=}0,p{=}2,q)
=fNormal(x;\mu,\sigma)=
| ||||||||||
v\sigma\sqrt{\pi |
v=\sqrt{2}
\sigma2
The Cauchy distribution has the pdf:
fSGT(x;\mu,\sigma,λ{=}0,p{=}2,q{=}\tfrac{1}{2})
=fCauchy(x;\mu,\sigma)=
1 | |||||
|
v=\sqrt{2}