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:

:
\begin
f(x;c,k) & = ck\frac \\ ptf(x;c,k,\lambda) & = \frac \left( \frac \right)^ \left + \left(\frac\right)^c\right
\end

The $\lambda$ parameter scales the underlying variate and is a positive real.

Cumulative distribution function

The cumulative distribution function is:

:$F(x;c,k) = 1-\left(1+x^c\right)^$
:F(x;c,k,\lambda) = 1 - \left + \left(\frac\right)^c \right

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 $U$ drawn from the uniform distribution in the interval $\left(0, 1\right)$, the random variable

:$X=\lambda \left (\frac-1 \right )^$

has a Burr Type XII distribution with parameters $c$, $k$ and $\lambda$. 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.See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr 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

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External links

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{{DEFAULTSORT:Burr Distribution
Continuous distributions
Systems of probability distributions