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Burr distribution
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<h2 id="definitions">Definitions</h2> <h3>Probability density function</h3> <p>The Burr (Type XII) distribution has <a href="/facts/Probability_density_function/zvfybna4">probability density function</a>:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> 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}}} <p>The λ {\displaystyle \lambda } parameter scales the underlying variate and is a positive real. </p> <h3>Cumulative distribution function</h3> <p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is: </p> 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}} <h2 id="applications">Applications</h2> <p>It is most commonly used to model <a href="/facts/Household_income/1AKezCVS">household income</a>, see for example: <a href="/facts/Household_income_in_the_United_States/LTyktYdQ">Household income in the U.S.</a> and compare to <a href="/facts/Magenta/dk2zD4hq">magenta</a> graph at right. </p> <h2 id="random-variate-generation">Random variate generation</h2> <p>Given a random variable U {\displaystyle U} drawn from the <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform distribution</a> in the interval ( 0 , 1 ) {\displaystyle \left(0,1\right)} , the random variable </p> X = λ ( 1 1 − U k − 1 ) 1 / c {\displaystyle X=\lambda \left({\frac {1}{\sqrt[{k}]{1-U}}}-1\right)^{1/c}} <p>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. </p> <h2 id="related-distributions">Related distributions</h2> <ul><li>When <i>c</i> = 1, the Burr distribution becomes the <a href="/facts/Lomax_distribution/ch9VY1mn">Lomax distribution</a>.</li></ul> <ul><li>When <i>k</i> = 1, the Burr distribution is a <a href="/facts/Log-logistic_distribution/ze7Pasb3">log-logistic distribution</a> sometimes referred to as the <a href="/facts/Fisk_distribution/ze7Pasb3">Fisk distribution</a>, a special case of the <a href="/facts/Champernowne_distribution/2rtxaalP">Champernowne distribution</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ul> <ul><li>The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li></ul> <ul><li>The <a href="/facts/Dagum_distribution/rmgoUXvM">Dagum distribution</a>, also known as the inverse Burr distribution, is the distribution of 1 / <i>X</i>, where <i>X</i> has the Burr distribution</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Rodriguez, R. N. (1977). <a href="http://www.lib.ncsu.edu/resolver/1840.4/3481">"A guide to Burr Type XII distributions"</a>. <i><a href="/facts/Biometrika/hg97ERcX">Biometrika</a></i>. 64 (1): 129–134. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fbiomet%2F64.1.129">10.1093/biomet/64.1.129</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li>John (2023-02-16). <a href="https://www.johndcook.com/blog/2023/02/15/all-burr-distributions/">"The other Burr distributions"</a>. <i>www.johndcook.com</i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756. <a href="https://doi.org/10.1214%2Faoms%2F1177731607" target="_blank">https://doi.org/10.1214%2Faoms%2F1177731607</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. doi:10.2307/1911538. JSTOR 1911538. <a href="/wiki/Econometrica" target="_blank">/wiki/Econometrica</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5. <a href="0-521-33825-5" target="_blank">0-521-33825-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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." <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644. <a href="/wiki/Econometrica" target="_blank">/wiki/Econometrica</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions." <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>