Paralogistic Distribution

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The paralogistic distribution is a sub-type of the generalized beta family of distributions. Introduced by McDonald in 1984 for applications in economics and actuarial science [1], it has since been applied to economic models of wealth and income [2].

Paralogistic distribution properties

The paralogistic family of distributions is created from the Burr distribution by setting the shape parameter, k, equal to 2.

The paralogistic distribution probability density function (PDF) is [2]

paralogistic function pdf


  • α = the shape parameter
  • θ = the scale parameter.

The scale parameter is sometimes represented as c instead of θ.

The 2-parameter paralogistic is equal to:

  • The 4-parameter generalized beta II distribution with shape parameter p = 1 and a = q [2]
  • The 3-parameter Singh-Maddala distribution with a = q [2]
  • The Burr distribution with γ = α [4].

The cumulative distribution function (CDF) is

One problem with the paralogistic distribution is that when the failure time doesn’t start at the origin, the failure start point is not usually known. This problem can be avoided by introducing a shift parameter to tell us the time of the first failure, such as the new three parameter paralogistic distribution (NTPLD) distribution seen in Idemudia, R. & Ekhosuehi’s paper [5], which has the PDF

new three parameter paralogistic distribution


  • 𝑐 > 0 is the scale parameter,
  • 𝛼 > 0 is the shape parameter and
  • 𝜃 > 0 is the location parameter.

If 𝜃 = 0, the NTPLD becomes the regular paralogistic PDF.


[1] Mc-Donald, J. B. (1984). Some generalized functions for the Size distributions in Economics and Actuarial Sciences, John Wiley and sons, New York.

[2] Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley and Sons, New York.

[3] R Documentation. Paralogistic {actuar} — The Paralogistic Dist. Retrieved July 27, 2023 from:

[4] Appendix A An Inventory of Continuous Distributions

[5] Idemudia, R. & Ekhosuehi, N. A NEW THREE-PARAMETER PARALOGISTIC : ITS PROPERTIES AND APPLICATION. Journal of Data Science,17(2). P. 239 – 258, 2019. DOI:10.6339/JDS.201904_17(2).0001

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