Burr distribution

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution^[1] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution^[2] and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".

Definitions

The Burr (Type XII) distribution has probability density function:^[3]^[4]

\begin{matrix} f (x; c, k) & = c k \frac{x^{c - 1}}{(1 + x^{c})^{k + 1}} \\ f (x; c, k, λ) & = \frac{c k}{λ} {(\frac{x}{λ})}^{c - 1} {[1 + {(\frac{x}{λ})}^{c}]}^{- k - 1} \end{matrix}

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

F (x; c, k) = 1 - {(1 + x^{c})}^{- k}

F (x; c, k, λ) = 1 - {[1 + {(\frac{x}{λ})}^{c}]}^{- k}

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.

Given a random variable $U$ drawn from the uniform distribution in the interval $(0, 1)$ , the random variable

X = λ {(\frac{1}{\sqrt[k]{1 - U}} - 1)}^{1 / c}

has a Burr Type XII distribution with parameters $c$ , $k$ and $λ$ . This follows from the inverse cumulative distribution function given above.

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.^[5]^[6]

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.^[7]

The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

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↑ Template:Cite book See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk Distribution."
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↑ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."