- Source: Burr distribution
In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution 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:
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
When c = 1, the Burr distribution becomes the Lomax distribution.
When k = 1, the Burr distribution is a log-logistic distribution sometimes referred to as the Fisk distribution, a special case of the Champernowne distribution.
The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.
The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution
References
Further reading
Rodriguez, R. N. (1977). "A guide to Burr Type XII distributions". Biometrika. 64 (1): 129–134. doi:10.1093/biomet/64.1.129.
External links
John (2023-02-16). "The other Burr distributions". www.johndcook.com.
Kata Kunci Pencarian:
- Bayam prancis
- Suku Dayak
- Timothée Chalamet
- Abad Pencerahan
- Homoseksualitas
- Chef (film)
- Daftar film Amerika tahun 2010
- The Federalist Papers
- Jaws
- Ikan dayung amerika
- Burr distribution
- Burr
- Probability distribution fitting
- Heavy-tailed distribution
- Generalized Pareto distribution
- Shape parameter
- Generalized beta distribution
- Relationships among probability distributions
- Burr Ridge, Illinois
- List of statistics articles
