User:Mathstat/Pareto generalizations

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The Generalized Pareto Distributions[edit]

Note: Most of the material below has been added to Pareto distribution. See the current revision of the article Pareto distribution and the Talk page Talk:Pareto distribution.

There is a hierarchy [1][2] of Pareto Distributions known as Pareto Type I, II, III, IV, and Feller-Pareto distributions.[3] Pareto Type IV contains Pareto Type I and II as special cases. The Feller-Pareto[4][2] distribution generalizes Pareto Type IV.

Pareto Types I-IV[edit]

The Pareto distribution hierarchy is summarized in the table comparing the survival distributions (complementary CDF). The Pareto distribution of the second kind is also known as the Lomax distribution,[5]

Pareto Distributions
Support Parameters
Type I
Type II
Type III
Type IV

The shape parameter α is the tail index, μ is location, xm is scale, 'γ is an inequality parameter. Some special cases of Pareto Type (IV) are:


Feller-Pareto distribution[edit]

Feller[6][2] defines a Pareto variable by transformation of a beta random variable Y, where the probability density function of Y is

where B( ) is the beta function.

When W has the Lomax distribution, and is a generalization of P(IV).


Existence of the mean, and variance depend on the tail index α (inequality index γ). In particular, fractional δ-moments exist for some δ>0, as shown in the table below, where δ is not necessarily an integer.

Moments of Pareto I-IV Distributions (case μ=0)
Condition Condition
Type I
Type II
Type III
Type IV


  1. ^ Arnold (1983), p. 45 (3.2.2).
  2. ^ a b c Johnson, Kotz, and Balakrishnan (1994), page 575, (20.4). Cite error: Invalid <ref> tag; name "jkb94" defined multiple times with different content (see the help page).
  3. ^ See Johnson, Kotz, and Balakrishnan (1994), Ch. 20, Arnold (1983), Ch. 3, and Kleiber and Kotz (2003), Ch. 3.
  4. ^ Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2 (Second edition), New York: Wiley.
  5. ^ Lomax, K. S. (1954). Business failures. Another example of the analysis of failure data. Journal of the American Statistical Association, 49, 847–852.
  6. ^ Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2 (Second edition), New York: Wiley.


  • N. L. Johnson, S. Kotz, and N. Balakrishnan (1994). Continuous Univariate Distributions Volume 1, Second Edition, Wiley.
  • Barry C. Arnold (2011). "Chapter 7: Pareto and Generalized Pareto Distributions". In Duangkamon Chotikapanich. Modeling Distributions and Lorenz Curves. New York: Springer.  Text "." ignored (help)
  • Arnold, B. C. and Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics.  Text "." ignored (help)