Family of continuous probability distributions
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]
Definition [ edit ]
Probability density function [ edit ]
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
,
q
)
=
p
2
v
σ しぐま
q
1
p
B
(
1
p
,
q
)
[
1
+
|
x
−
μ みゅー
+
m
|
p
q
(
v
σ しぐま
)
p
(
1
+
λ らむだ
sgn
(
x
−
μ みゅー
+
m
)
)
p
]
1
p
+
q
{\displaystyle f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p,q)={\frac {p}{2v\sigma q^{\frac {1}{p}}B({\frac {1}{p}},q)\left[1+{\frac {|x-\mu +m|^{p}}{q(v\sigma )^{p}(1+\lambda \operatorname {sgn}(x-\mu +m))^{p}}}\right]^{{\frac {1}{p}}+q}}}}
where
B
{\displaystyle B}
is the beta function ,
μ みゅー
{\displaystyle \mu }
is the location parameter,
σ しぐま
>
0
{\displaystyle \sigma >0}
is the scale parameter,
−
1
<
λ らむだ
<
1
{\displaystyle -1<\lambda <1}
is the skewness parameter, and
p
>
0
{\displaystyle p>0}
and
q
>
0
{\displaystyle q>0}
are the parameters that control the kurtosis .
m
{\displaystyle m}
and
v
{\displaystyle 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
=
λ らむだ
v
σ しぐま
2
q
1
p
B
(
2
p
,
q
−
1
p
)
B
(
1
p
,
q
)
{\displaystyle m=\lambda v\sigma {\frac {2q^{\frac {1}{p}}B({\frac {2}{p}},q-{\frac {1}{p}})}{B({\frac {1}{p}},q)}}}
and
v
=
q
−
1
p
(
1
+
3
λ らむだ
2
)
B
(
3
p
,
q
−
2
p
)
B
(
1
p
,
q
)
−
4
λ らむだ
2
B
(
2
p
,
q
−
1
p
)
2
B
(
1
p
,
q
)
2
{\displaystyle v={\frac {q^{-{\frac {1}{p}}}}{\sqrt {(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}}}}}}}
.
These values for
m
{\displaystyle m}
and
v
{\displaystyle v}
yield a distribution with mean of
μ みゅー
{\displaystyle \mu }
if
p
q
>
1
{\displaystyle pq>1}
and a variance of
σ しぐま
2
{\displaystyle \sigma ^{2}}
if
p
q
>
2
{\displaystyle pq>2}
. In order for
m
{\displaystyle m}
to take on this value however, it must be the case that
p
q
>
1
{\displaystyle pq>1}
. Similarly, for
v
{\displaystyle v}
to equal the above value,
p
q
>
2
{\displaystyle pq>2}
.
The parameterization that yields the simplest functional form of the probability density function sets
m
=
0
{\displaystyle m=0}
and
v
=
1
{\displaystyle v=1}
. This gives a mean of
μ みゅー
+
2
v
σ しぐま
λ らむだ
q
1
p
B
(
2
p
,
q
−
1
p
)
B
(
1
p
,
q
)
{\displaystyle \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
σ しぐま
2
q
2
p
(
(
1
+
3
λ らむだ
2
)
B
(
3
p
,
q
−
2
p
)
B
(
1
p
,
q
)
−
4
λ らむだ
2
B
(
2
p
,
q
−
1
p
)
2
B
(
1
p
,
q
)
2
)
{\displaystyle \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
λ らむだ
{\displaystyle \lambda }
parameter controls the skewness of the distribution. To see this, let
M
{\displaystyle M}
denote the mode of the distribution, and
∫
−
∞
M
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
,
q
)
d
x
=
1
−
λ らむだ
2
{\displaystyle \int _{-\infty }^{M}f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p,q)\mathrm {d} x={\frac {1-\lambda }{2}}}
Since
−
1
<
λ らむだ
<
1
{\displaystyle -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
λ らむだ
{\displaystyle \lambda }
. Thus the skewed generalized t distribution can be highly skewed as well as symmetric. If
−
1
<
λ らむだ
<
0
{\displaystyle -1<\lambda <0}
, then the distribution is negatively skewed. If
0
<
λ らむだ
<
1
{\displaystyle 0<\lambda <1}
, then the distribution is positively skewed. If
λ らむだ
=
0
{\displaystyle \lambda =0}
, then the distribution is symmetric.
Finally,
p
{\displaystyle p}
and
q
{\displaystyle q}
control the kurtosis of the distribution. As
p
{\displaystyle p}
and
q
{\displaystyle q}
get smaller, the kurtosis increases[1] (i.e. becomes more leptokurtic). Large values of
p
{\displaystyle p}
and
q
{\displaystyle q}
yield a distribution that is more platykurtic.
Moments [ edit ]
Let
X
{\displaystyle X}
be a random variable distributed with the skewed generalized t distribution. The
h
t
h
{\displaystyle h^{th}}
moment (i.e.
E
[
(
X
−
E
(
X
)
)
h
]
{\displaystyle E[(X-E(X))^{h}]}
), for
p
q
>
h
{\displaystyle pq>h}
, is:
∑
r
=
0
h
(
h
r
)
(
(
1
+
λ らむだ
)
r
+
1
+
(
−
1
)
r
(
1
−
λ らむだ
)
r
+
1
)
(
−
λ らむだ
)
h
−
r
(
v
σ しぐま
)
h
q
h
p
B
(
r
+
1
p
,
q
−
r
p
)
B
(
2
p
,
q
−
1
p
)
h
−
r
2
r
−
h
+
1
B
(
1
p
,
q
)
h
−
r
+
1
{\displaystyle \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
p
q
>
1
{\displaystyle pq>1}
, is:
μ みゅー
+
2
v
σ しぐま
λ らむだ
q
1
p
B
(
2
p
,
q
−
1
p
)
B
(
1
p
,
q
)
−
m
{\displaystyle \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
]
{\displaystyle E[(X-E(X))^{2}]}
), for
p
q
>
2
{\displaystyle pq>2}
, is:
(
v
σ しぐま
)
2
q
2
p
(
(
1
+
3
λ らむだ
2
)
B
(
3
p
,
q
−
2
p
)
B
(
1
p
,
q
)
−
4
λ らむだ
2
B
(
2
p
,
q
−
1
p
)
2
B
(
1
p
,
q
)
2
)
{\displaystyle (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
]
{\displaystyle E[(X-E(X))^{3}]}
), for
p
q
>
3
{\displaystyle pq>3}
, is:
2
q
3
/
p
λ らむだ
(
v
σ しぐま
)
3
B
(
1
p
,
q
)
3
(
8
λ らむだ
2
B
(
2
p
,
q
−
1
p
)
3
−
3
(
1
+
3
λ らむだ
2
)
B
(
1
p
,
q
)
{\displaystyle {\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)}
×
B
(
2
p
,
q
−
1
p
)
B
(
3
p
,
q
−
2
p
)
+
2
(
1
+
λ らむだ
2
)
B
(
1
p
,
q
)
2
B
(
4
p
,
q
−
3
p
)
)
{\displaystyle \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
]
{\displaystyle E[(X-E(X))^{4}]}
), for
p
q
>
4
{\displaystyle pq>4}
, is:
q
4
/
p
(
v
σ しぐま
)
4
B
(
1
p
,
q
)
4
(
−
48
λ らむだ
4
B
(
2
p
,
q
−
1
p
)
4
+
24
λ らむだ
2
(
1
+
3
λ らむだ
2
)
B
(
1
p
,
q
)
B
(
2
p
,
q
−
1
p
)
2
{\displaystyle {\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}}
×
B
(
3
p
,
q
−
2
p
)
−
32
λ らむだ
2
(
1
+
λ らむだ
2
)
B
(
1
p
,
q
)
2
B
(
2
p
,
q
−
1
p
)
B
(
4
p
,
q
−
3
p
)
{\displaystyle \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
λ らむだ
2
+
5
λ らむだ
4
)
B
(
1
p
,
q
)
3
B
(
5
p
,
q
−
4
p
)
)
{\displaystyle +(1+10\lambda ^{2}+5\lambda ^{4})B({\frac {1}{p}},q)^{3}B({\frac {5}{p}},q-{\frac {4}{p}}){\Bigg )}}
Special Cases [ edit ]
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 [ edit ]
The Skewed Generalized Error Distribution (SGED) has the pdf:
lim
q
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
,
q
)
{\displaystyle \lim _{q\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p,q)}
=
f
SGED
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
)
=
p
2
v
σ しぐま
Γ がんま
(
1
p
)
e
−
(
|
x
−
μ みゅー
+
m
|
v
σ しぐま
[
1
+
λ らむだ
sgn
(
x
−
μ みゅー
+
m
)
]
)
p
{\displaystyle =f_{\text{SGED}}(x;\mu ,\sigma ,\lambda ,p)={\frac {p}{2v\sigma \Gamma ({\frac {1}{p}})}}e^{-\left({\frac {|x-\mu +m|}{v\sigma [1+\lambda \operatorname {sgn}(x-\mu +m)]}}\right)^{p}}}
where
m
=
λ らむだ
v
σ しぐま
2
2
p
Γ がんま
(
1
2
+
1
p
)
π ぱい
{\displaystyle m=\lambda v\sigma {\frac {2^{\frac {2}{p}}\Gamma ({\frac {1}{2}}+{\frac {1}{p}})}{\sqrt {\pi }}}}
gives a mean of
μ みゅー
{\displaystyle \mu }
. Also
v
=
π ぱい
Γ がんま
(
1
p
)
π ぱい
(
1
+
3
λ らむだ
2
)
Γ がんま
(
3
p
)
−
16
1
p
λ らむだ
2
Γ がんま
(
1
2
+
1
p
)
2
Γ がんま
(
1
p
)
{\displaystyle 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
σ しぐま
2
{\displaystyle \sigma ^{2}}
.
Generalized t -distribution [ edit ]
The generalized t -distribution (GT) has the pdf:
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
=
0
,
p
,
q
)
{\displaystyle f_{\text{SGT}}(x;\mu ,\sigma ,\lambda {=}0,p,q)}
=
f
GT
(
x
;
μ みゅー
,
σ しぐま
,
p
,
q
)
=
p
2
v
σ しぐま
q
1
p
B
(
1
p
,
q
)
[
1
+
|
x
−
μ みゅー
|
p
q
(
v
σ しぐま
)
p
]
1
p
+
q
{\displaystyle =f_{\text{GT}}(x;\mu ,\sigma ,p,q)={\frac {p}{2v\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
=
1
q
1
p
B
(
1
p
,
q
)
B
(
3
p
,
q
−
2
p
)
{\displaystyle 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
σ しぐま
2
{\displaystyle \sigma ^{2}}
.
Skewed t -distribution [ edit ]
The skewed t -distribution (ST) has the pdf:
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
=
2
,
q
)
{\displaystyle f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p{=}2,q)}
=
f
ST
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
q
)
=
Γ がんま
(
1
2
+
q
)
v
σ しぐま
(
π ぱい
q
)
1
2
Γ がんま
(
q
)
[
1
+
|
x
−
μ みゅー
+
m
|
2
q
(
v
σ しぐま
)
2
(
1
+
λ らむだ
sgn
(
x
−
μ みゅー
+
m
)
)
2
]
1
2
+
q
{\displaystyle =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 \operatorname {sgn}(x-\mu +m))^{2}}}\right]^{{\frac {1}{2}}+q}}}}
where
m
=
λ らむだ
v
σ しぐま
2
q
1
2
Γ がんま
(
q
−
1
2
)
π ぱい
1
2
Γ がんま
(
q
)
{\displaystyle m=\lambda v\sigma {\frac {2q^{\frac {1}{2}}\Gamma (q-{\frac {1}{2}})}{\pi ^{\frac {1}{2}}\Gamma (q)}}}
gives a mean of
μ みゅー
{\displaystyle \mu }
. Also
v
=
1
q
1
2
(
1
+
3
λ らむだ
2
)
1
2
q
−
2
−
4
λ らむだ
2
π ぱい
(
Γ がんま
(
q
−
1
2
)
Γ がんま
(
q
)
)
2
{\displaystyle 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
σ しぐま
2
{\displaystyle \sigma ^{2}}
.
Skewed Laplace distribution [ edit ]
The skewed Laplace distribution (SLaplace) has the pdf:
lim
q
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
=
1
,
q
)
{\displaystyle \lim _{q\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p{=}1,q)}
=
f
SLaplace
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
)
=
1
2
v
σ しぐま
e
−
|
x
−
μ みゅー
+
m
|
v
σ しぐま
(
1
+
λ らむだ
sgn
(
x
−
μ みゅー
+
m
)
)
{\displaystyle =f_{\text{SLaplace}}(x;\mu ,\sigma ,\lambda )={\frac {1}{2v\sigma }}e^{-{\frac {|x-\mu +m|}{v\sigma (1+\lambda \operatorname {sgn}(x-\mu +m))}}}}
where
m
=
2
v
σ しぐま
λ らむだ
{\displaystyle m=2v\sigma \lambda }
gives a mean of
μ みゅー
{\displaystyle \mu }
. Also
v
=
[
2
(
1
+
λ らむだ
2
)
]
−
1
2
{\displaystyle v=[2(1+\lambda ^{2})]^{-{\frac {1}{2}}}}
gives a variance of
σ しぐま
2
{\displaystyle \sigma ^{2}}
.
Generalized error distribution [ edit ]
The generalized error distribution (GED, also known as the generalized normal distribution ) has the pdf:
lim
q
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
=
0
,
p
,
q
)
{\displaystyle \lim _{q\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda {=}0,p,q)}
=
f
GED
(
x
;
μ みゅー
,
σ しぐま
,
p
)
=
p
2
v
σ しぐま
Γ がんま
(
1
p
)
e
−
(
|
x
−
μ みゅー
|
v
σ しぐま
)
p
{\displaystyle =f_{\text{GED}}(x;\mu ,\sigma ,p)={\frac {p}{2v\sigma \Gamma ({\frac {1}{p}})}}e^{-\left({\frac {|x-\mu |}{v\sigma }}\right)^{p}}}
where
v
=
Γ がんま
(
1
p
)
Γ がんま
(
3
p
)
{\displaystyle v={\sqrt {\frac {\Gamma ({\frac {1}{p}})}{\Gamma ({\frac {3}{p}})}}}}
gives a variance of
σ しぐま
2
{\displaystyle \sigma ^{2}}
.
Skewed normal distribution [ edit ]
The skewed normal distribution (SNormal) has the pdf:
lim
q
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
=
2
,
q
)
{\displaystyle \lim _{q\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p{=}2,q)}
=
f
SNormal
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
)
=
1
v
σ しぐま
π ぱい
e
−
[
|
x
−
μ みゅー
+
m
|
v
σ しぐま
(
1
+
λ らむだ
sgn
(
x
−
μ みゅー
+
m
)
)
]
2
{\displaystyle =f_{\text{SNormal}}(x;\mu ,\sigma ,\lambda )={\frac {1}{v\sigma {\sqrt {\pi }}}}e^{-\left[{\frac {|x-\mu +m|}{v\sigma (1+\lambda \operatorname {sgn}(x-\mu +m))}}\right]^{2}}}
where
m
=
λ らむだ
v
σ しぐま
2
π ぱい
{\displaystyle m=\lambda v\sigma {\frac {2}{\sqrt {\pi }}}}
gives a mean of
μ みゅー
{\displaystyle \mu }
. Also
v
=
2
π ぱい
π ぱい
−
8
λ らむだ
2
+
3
π ぱい
λ らむだ
2
{\displaystyle v={\sqrt {\frac {2\pi }{\pi -8\lambda ^{2}+3\pi \lambda ^{2}}}}}
gives a variance of
σ しぐま
2
{\displaystyle \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
−
λ らむだ
{\displaystyle 1-\lambda }
and
1
+
λ らむだ
{\displaystyle 1+\lambda }
.
Student's t -distribution [ edit ]
The Student's t-distribution (T) has the pdf:
f
SGT
(
x
;
μ みゅー
=
0
,
σ しぐま
=
1
,
λ らむだ
=
0
,
p
=
2
,
q
=
d
2
)
{\displaystyle f_{\text{SGT}}(x;\mu {=}0,\sigma {=}1,\lambda {=}0,p{=}2,q{=}{\tfrac {d}{2}})}
=
f
T
(
x
;
d
)
=
Γ がんま
(
d
+
1
2
)
(
π ぱい
d
)
1
2
Γ がんま
(
d
2
)
(
1
+
x
2
d
)
−
d
+
1
2
{\displaystyle =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
=
2
{\displaystyle v={\sqrt {2}}}
was substituted.
Skewed Cauchy distribution [ edit ]
The skewed cauchy distribution (SCauchy) has the pdf:
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
=
2
,
q
=
1
2
)
{\displaystyle f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p{=}2,q{=}{\tfrac {1}{2}})}
=
f
SCauchy
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
)
=
1
σ しぐま
π ぱい
[
1
+
|
x
−
μ みゅー
|
2
σ しぐま
2
(
1
+
λ らむだ
sgn
(
x
−
μ みゅー
)
)
2
]
{\displaystyle =f_{\text{SCauchy}}(x;\mu ,\sigma ,\lambda )={\frac {1}{\sigma \pi \left[1+{\frac {\left|x-\mu \right|^{2}}{\sigma ^{2}(1+\lambda \operatorname {sgn}(x-\mu ))^{2}}}\right]}}}
v
=
2
{\displaystyle v={\sqrt {2}}}
and
m
=
0
{\displaystyle m=0}
was substituted.
The mean, variance, skewness, and kurtosis of the skewed Cauchy distribution are all undefined.
Laplace distribution [ edit ]
The Laplace distribution has the pdf:
lim
q
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
=
0
,
p
=
1
,
q
)
{\displaystyle \lim _{q\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda {=}0,p{=}1,q)}
=
f
Laplace
(
x
;
μ みゅー
,
σ しぐま
)
=
1
2
σ しぐま
e
−
|
x
−
μ みゅー
|
σ しぐま
{\displaystyle =f_{\text{Laplace}}(x;\mu ,\sigma )={\frac {1}{2\sigma }}e^{-{\frac {|x-\mu |}{\sigma }}}}
v
=
1
{\displaystyle v=1}
was substituted.
Uniform Distribution [ edit ]
The uniform distribution has the pdf:
lim
p
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
,
p
,
q
)
{\displaystyle \lim _{p\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda ,p,q)}
=
f
(
x
)
=
{
1
2
v
σ しぐま
|
x
−
μ みゅー
|
<
v
σ しぐま
0
o
t
h
e
r
w
i
s
e
{\displaystyle =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
μ みゅー
=
a
+
b
2
{\displaystyle \mu ={\frac {a+b}{2}}}
,
v
=
1
{\displaystyle v=1}
, and
σ しぐま
=
b
−
a
2
{\displaystyle \sigma ={\frac {b-a}{2}}}
.
Normal distribution [ edit ]
The normal distribution has the pdf:
lim
q
→
∞
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
=
0
,
p
=
2
,
q
)
{\displaystyle \lim _{q\to \infty }f_{\text{SGT}}(x;\mu ,\sigma ,\lambda {=}0,p{=}2,q)}
=
f
Normal
(
x
;
μ みゅー
,
σ しぐま
)
=
e
−
(
|
x
−
μ みゅー
|
v
σ しぐま
)
2
v
σ しぐま
π ぱい
{\displaystyle =f_{\text{Normal}}(x;\mu ,\sigma )={\frac {e^{-\left({\frac {|x-\mu |}{v\sigma }}\right)^{2}}}{v\sigma {\sqrt {\pi }}}}}
where
v
=
2
{\displaystyle v={\sqrt {2}}}
gives a variance of
σ しぐま
2
{\displaystyle \sigma ^{2}}
.
Cauchy Distribution [ edit ]
The Cauchy distribution has the pdf:
f
SGT
(
x
;
μ みゅー
,
σ しぐま
,
λ らむだ
=
0
,
p
=
2
,
q
=
1
2
)
{\displaystyle f_{\text{SGT}}(x;\mu ,\sigma ,\lambda {=}0,p{=}2,q{=}{\tfrac {1}{2}})}
=
f
Cauchy
(
x
;
μ みゅー
,
σ しぐま
)
=
1
σ しぐま
π ぱい
[
1
+
(
x
−
μ みゅー
σ しぐま
)
2
]
{\displaystyle =f_{\text{Cauchy}}(x;\mu ,\sigma )={\frac {1}{\sigma \pi \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{2}\right]}}}
v
=
2
{\displaystyle v={\sqrt {2}}}
was substituted.
References [ edit ]
Hansen, B. (1994). "Autoregressive Conditional Density Estimation". International Economic Review . 35 (3): 705–730. doi :10.2307/2527081 . JSTOR 2527081 .
Hansen, C.; McDonald, J.; Newey, W. (2010). "Instrumental Variables Estimation with Flexible Distributions". Journal of Business and Economic Statistics . 28 : 13–25. doi :10.1198/jbes.2009.06161 . hdl :10419/79273 . S2CID 11370711 .
Hansen, C.; McDonald, J.; Theodossiou, P. (2007). "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models" . Economics: The Open-Access, Open-Assessment e-Journal . 1 (2007–7): 1. doi :10.5018/economics-ejournal.ja.2007-7 . hdl :20.500.14279/1024 .
McDonald, J.; Michefelder, R.; Theodossiou, P. (2009). "Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application" (PDF) . Multinational Finance Journal . 15 (3/4): 293–321. doi :10.17578/13-3/4-6 . S2CID 15012865 .
McDonald, J.; Michelfelder, R.; Theodossiou, P. (2010). "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas". Quantitative Finance . 10 (4): 375–387. doi :10.1080/14697680902814241 . S2CID 11130911 .
McDonald, J.; Newey, W. (1988). "Partially Adaptive Estimation of Regression Models via the Generalized t Distribution". Econometric Theory . 4 (3): 428–457. doi :10.1017/s0266466600013384 . S2CID 120305707 .
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 (12–part–1): 1650–1661. doi :10.1287/mnsc.44.12.1650 .
External links [ edit ]
^ a b c d Theodossiou, P (1998). "Financial Data and the Skewed Generalized T Distribution". Management Science . 44 (12–part–1): 1650–1661. doi :10.1287/mnsc.44.12.1650 .
^ a b Hansen, C.; McDonald, J.; Newey, W. (2010). "Instrumental Variables Estimation with Flexible Distributions". Journal of Business and Economic Statistics . 28 : 13–25. doi :10.1198/jbes.2009.06161 . hdl :10419/79273 . S2CID 11370711 .
^ 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
^ McDonald, J.; Michelfelder, R.; Theodossiou, P. (2009). "Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application" (PDF) . Multinational Finance Journal . 15 (3/4): 293–321. doi :10.17578/13-3/4-6 . S2CID 15012865 .
^ a b McDonald J., R. Michelfelder, and P. Theodossiou (2010) "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas" Quantitative Finance 375-387.
^ a b McDonald, J.; Newey, W. (1998). "Partially Adaptive Estimation of Regression Models via the Generalized t Distribution". Econometric Theory . 4 (3): 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.
^ Hansen, B (1994). "Autoregressive Conditional Density Estimation". International Economic Review . 35 (3): 705–730. doi :10.2307/2527081 . JSTOR 2527081 .
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families