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Burr Type XII
Probability density function
Cumulative distribution function
Parameters
c
>
0
{\displaystyle c>0\!}
k
>
0
{\displaystyle k>0\!}
Support
x
>
0
{\displaystyle x>0\!}
PDF
c
k
x
c
−
1
(
1
+
x
c
)
k
+
1
{\displaystyle ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\!}
CDF
1
−
(
1
+
x
c
)
−
k
{\displaystyle 1-\left(1+x^{c}\right)^{-k}}
Mean
μ
1
=
k
B
(
k
−
1
/
c
,
1
+
1
/
c
)
{\displaystyle \mu _{1}=k\operatorname {\mathrm {B} } (k-1/c,\,1+1/c)}
where Β() is the
beta function
Median
(
2
1
k
−
1
)
1
c
{\displaystyle \left(2^{\frac {1}{k}}-1\right)^{\frac {1}{c}}}
Mode
(
c
−
1
k
c
+
1
)
1
c
{\displaystyle \left({\frac {c-1}{kc+1}}\right)^{\frac {1}{c}}}
Variance
−
μ
1
2
+
μ
2
{\displaystyle -\mu _{1}^{2}+\mu _{2}}
Skewness
2
μ
1
3
−
3
μ
1
μ
2
+
μ
3
(
−
μ
1
2
+
μ
2
)
3
/
2
{\displaystyle {\frac {2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{3/2}}}}
Excess kurtosis
−
3
μ
1
4
+
6
μ
1
2
μ
2
−
4
μ
1
μ
3
+
μ
4
(
−
μ
1
2
+
μ
2
)
2
−
3
{\displaystyle {\frac {-3\mu _{1}^{4}+6\mu _{1}^{2}\mu _{2}-4\mu _{1}\mu _{3}+\mu _{4}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{2}}}-3}
where moments (
see )
μ
r
=
k
B
(
c
k
−
r
c
,
c
+
r
c
)
{\displaystyle \mu _{r}=k\operatorname {\mathrm {B} } \left({\frac {ck-r}{c}},\,{\frac {c+r}{c}}\right)}
CF
=
c
(
−
i
t
)
k
c
Γ
(
k
)
H
1
,
2
2
,
1
(
−
i
t
)
c
|
(
−
k
,
1
)
(
0
,
1
)
,
(
−
k
c
,
c
)
,
t
≠
0
{\displaystyle ={\frac {c(-it)^{kc}}{\Gamma (k)}}H_{1,2}^{2,1}\!\left[(-it)^{c}\left|{\begin{matrix}(-k,1)\\(0,1),(-kc,c)\end{matrix}}\right.\right],t\neq 0}
=
1
,
t
=
0
{\displaystyle =1,t=0}
where
Γ
{\displaystyle \Gamma }
is the
Gamma function and
H
{\displaystyle H}
is the
Fox H-function .
[1]
In
probability theory ,
statistics and
econometrics , the Burr Type XII distribution or simply the Burr distribution
[2] is a
continuous probability distribution for a non-negative
random variable . It is also known as the Singh–Maddala distribution
[3] and is one of a number of different distributions sometimes called the "generalized
log-logistic distribution ".
Definitions
Probability density function
The Burr (Type XII) distribution has
probability density function :
[4]
[5]
f
(
x
;
c
,
k
)
=
c
k
x
c
−
1
(
1
+
x
c
)
k
+
1
f
(
x
;
c
,
k
,
λ
)
=
c
k
λ
(
x
λ
)
c
−
1
1
+
(
x
λ
)
c
−
k
−
1
{\displaystyle {\begin{aligned}f(x;c,k)&=ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\\[6pt]f(x;c,k,\lambda )&={\frac {ck}{\lambda }}\left({\frac {x}{\lambda }}\right)^{c-1}\left[1+\left({\frac {x}{\lambda }}\right)^{c}\right]^{-k-1}\end{aligned}}}
The
λ
{\displaystyle \lambda }
parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The
cumulative distribution function is:
F
(
x
;
c
,
k
)
=
1
−
(
1
+
x
c
)
−
k
{\displaystyle F(x;c,k)=1-\left(1+x^{c}\right)^{-k}}
F
(
x
;
c
,
k
,
λ
)
=
1
−
1
+
(
x
λ
)
c
−
k
{\displaystyle F(x;c,k,\lambda )=1-\left[1+\left({\frac {x}{\lambda }}\right)^{c}\right]^{-k}}
Applications
It is most commonly used to model
household income , see for example:
Household income in the U.S. and compare to
magenta graph at right.
Random variate generation
Given a random variable
U
{\displaystyle U}
drawn from the
uniform distribution in the interval
(
0
,
1
)
{\displaystyle \left(0,1\right)}
, the random variable
X
=
λ
(
1
1
−
U
k
−
1
)
1
/
c
{\displaystyle X=\lambda \left({\frac {1}{\sqrt[{k}]{1-U}}}-1\right)^{1/c}}
has a Burr Type XII distribution with parameters
c
{\displaystyle c}
,
k
{\displaystyle k}
and
λ
{\displaystyle \lambda }
. This follows from the inverse cumulative distribution function given above.
Related distributions
The Burr Type XII distribution is a member of a system of continuous distributions introduced by
Irving W. Burr (1942), which comprises 12 distributions.
[8]
The
Dagum distribution , also known as the inverse Burr distribution, is the distribution of 1 / X , where X has the Burr distribution
References
^ Nadarajah, S.; Pogány, T. K.; Saxena, R. K. (2012). "On the characteristic function for Burr distributions". Statistics . 46 (3): 419–428.
doi :
10.1080/02331888.2010.513442 .
S2CID
120848446 .
^ Burr, I. W. (1942).
"Cumulative frequency functions" .
Annals of Mathematical Statistics . 13 (2): 215–232.
doi :
10.1214/aoms/1177731607 .
JSTOR
2235756 .
^ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes".
Econometrica . 44 (5): 963–970.
doi :
10.2307/1911538 .
JSTOR
1911538 .
^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics . Cambridge University Press.
ISBN
0-521-33825-5 .
^ Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", International Statistical Review , 48 (3): 337–344,
doi :
10.2307/1402945 ,
JSTOR
1402945
^ C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences . New York: Wiley. See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk Distribution."
^ Champernowne, D. G. (1952). "The graduation of income distributions".
Econometrica . 20 (4): 591–614.
doi :
10.2307/1907644 .
JSTOR
1907644 .
^ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."
Further reading
External links
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