Burr distribution explained

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

Probability density function

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

\begin{align} f(x;c,k) & = ck (x ^{c-1})/(( 1+ x ^{c)} ^{k+1}) \\[ 6 pt] f(x;c,k, λ ) & = (ck)/(λ) \left( (x)/(λ) \right) ^{c-1} \left[ 1 + \left( (x)/(λ) \right) ^{c\right]} ^{-k-1} \end{align}

The

λ 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)} ^{-k} F(x;c,k, λ ) = 1 - \left[ 1 + \left( (x)/(λ) \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 drawn from the uniform distribution in the interval \left( 0 , 1 \right) , the random variableX =λ \left ( (1)/(\sqrt[k]{ 1- U) }-1 \right)^

has a Burr Type XII distribution with parameters

c , k and λ . 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.^{[4] [5]}
• The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.^[6]
• The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

Further reading

• Rodriguez . R. N. . 1977 . A guide to Burr Type XII distributions . . 64 . 1 . 129–134 . 10.1093/biomet/64.1.129 .

External links

• Web site: John . 2023-02-16 . The other Burr distributions . www.johndcook.com . en-US.

Notes and References

1. Burr . I. W. . 1942 . Cumulative frequency functions . . 13 . 2 . 215–232 . 2235756 . 10.1214/aoms/1177731607. free .
2. Maddala . G. . Singh . S. . 1976 . A Function for the Size Distribution of Incomes . . 44 . 5 . 963–970 . 1911538 . 10.2307/1911538 .
3. Book: Maddala, G. S. . 1983 . 1996 . Limited-Dependent and Qualitative Variables in Econometrics . Cambridge University Press . 0-521-33825-5 .
4. Book: C. Kleiber and S. Kotz. Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. New York. 2003. See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk Distribution."
5. Champernowne . D. G.. . The graduation of income distributions . 1952 . 20 . 4 . 591–614 . 10.2307/1907644. 1907644.
6. See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."