Favorable Estimators for Fitting Pareto Models: A Study Using Goodness-of-fit Measures with Actual Data

Abstract

Several recent papers treated robust and efficient estimation of tail index parameters for (equivalent) Pareto and truncated exponential models, for large and small samples. New robust estimators of "generalized median" (GM) and "trimmed mean" (T) type were introduced and shown to provide more favorable trade-offs between efficiency and robustness than several well-established estimators, including those corresponding to methods of maximum likelihood, quantiles, and percentile matching. Here we investigate performance of the above mentioned estimators on real data and establish -via the use of goodness-of-fit measures -that favorable theoretical properties of the GM and T type estimators translate into an excellent practical performance. Further, we arrive at guidelines for Pareto model diagnostics, testing, and selection of particular robust estimators in practice. Model fits provided by the estimators are ranked and compared on the basis of Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling statistics.

On Sums of Claims and their Applications in Analysis of Pension Funds and Insurance Products

“…Our fi ndings for the Pareto model are in accord with conclusions of Brazauskas and Serfl ing (2003) that the maximum likelihood (ML) estimator is effi cient but not robust and should be replaced by a competitor. It is not robust also for the case of mispecifi cation of the heavytailed distribution.…”

Modelling Insurance Data With the Pareto Arctan Distribution

“…In the following, two more measures of model assessment for individual data based on the empirical distribution function such as the Kolmogorov-Smirnov (KS) test and Crámer von Mises (CvM) test (see Brazauskas and Serfling (2003) and Rizzo (2009) for details) have been applied to these six models. For these tests, smaller values indicate a better fit of the distribution to the data.…”

“…Appropriate estimation of these extreme events is crucial for the practitioner to correctly assess insurance and reinsurance premiums. On this subject, the single parameter Pareto distribution (Arnold (1983), Brazauskas and Serfling (2003), Rytgaard (1990), among others) has been traditionally considered as a suitable claim size distribution in relation to rating problems. Concerning this, the single parameter Pareto distribution, apart from its nice properties, provides a good depiction of the random behavior of large losses (e.g.…”

Regarding folded models and the paper by Brazauskas and Kleefeld (2011)

“…This may be due to one or more of round-off and other forms of numerical error, different parameter estimation routines, etc. Also, I 'de-grouped' the Norwegian claims data in order to remove ties and avoid inappropriate clustering of claims due to rounding using a method similar to that described in Brazauskas and Serfling (2003), and this de-grouping may or may not have been done in Brazauskas and Kleefeld (2011). My maximum likelihood-based parameter estimates were all determined using the nlm function in R. For the record, the parameter estimates determined werer ¼ 1:157 (for the t 7 ) andr ¼ 1:171 andĉ ¼ (0:125 (for the GPD).…”