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.
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
Lomax
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:
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.
^ abcJohnson, Kotz, and Balakrishnan (1994), page 575, (20.4). Cite error: The named reference "jkb94" was defined multiple times with different content (see the help page).
^See Johnson, Kotz, and Balakrishnan (1994), Ch. 20, Arnold (1983), Ch. 3, and Kleiber and Kotz (2003), Ch. 3.
^Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2 (Second edition), New York: Wiley.
^Lomax, K. S. (1954). Business failures. Another example of the analysis of failure data. Journal of the American Statistical Association, 49, 847–852.
^Feller, W. (1971). An Introduction to Probability Theory and its Applications, 2 (Second edition), New York: Wiley.
Arnold, B. C. and Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics. {{cite book}}: Text "." ignored (help)CS1 maint: multiple names: authors list (link)