Generalizing the Pareto to the log-Pareto model and statistical inference

• Published in: Extremes (Boston) Vol. 12; no. 1; pp. 93 - 105
• Main Authors: Cormann, Ulf, Reiss, Rolf-Dieter
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2009; Springer Nature B.V
• Subjects: Civil Engineering; Economics; Environmental Management; Estimating techniques; Finance; Hierarchies; Hydrogeology; Insurance; Management; Mathematics and Statistics; Pareto optimum; Quality Control; Random variables; Reliability; Safety and Risk; Statistical inference; Statistics; Statistics for Business; Studies; Pareto; Copepod data; Quick estimators; Exceedances; Log-Pareto and generalized log-Pareto dfs; MLE; P-max and p-pot stable dfs; Super-heavy tails; Primary—62F03, 62F10, 62G43
• ISSN: 1386-1999; 1572-915X
• DOI: 10.1007/s10687-008-0070-6

In this article we introduce a full-fledged statistical model of log-Pareto distribution functions (dfs) parametrized by two shape parameters and a scale parameter. Pareto dfs can be regained in the limit by varying parameters of log-Pareto dfs, whence the log-Pareto model can be regarded as an extension of the Pareto model. Log-Pareto dfs are first of all obtained by means of exponential transformations of Pareto dfs. We also indicate an iterated application of such a procedure. A class of generalized log-Pareto dfs is considered as well. In addition, power-pot (p-pot) stable dfs – related to p-max stable dfs – are introduced and log-Pareto dfs are identified as special cases. A modification of a quick (systematic) estimator is proposed as an initial estimator for the numerical computation of the maximum likelihood estimator (MLE) in the 3-parameter model.