Abstract
The Pareto distribution is often used in many areas of economics to model the right tail of heavy-tailed distributions. However, the standard method of estimating the shape parameter (the Pareto tail index) of this distribution—the maximum likelihood estimator (MLE), also known as the Hill estimator—is non-robust, in the sense that it is very sensitive to extreme observations, data contamination or model deviation. In recent years, a number of robust estimators for the Pareto tail index have been proposed, which correct the deficiency of the MLE. However, little is known about the performance of these estimators in small-sample setting, which often occurs in practice. This paper investigates the small-sample properties of the most popular robust estimators for the Pareto tail index, including the optimal B-robust estimator (Victoria-Feser and Ronchetti in Can J Stat 22:247–258, 1994), the weighted maximum likelihood estimator (Dupuis and Victoria-Feser in Can J Stat 34:639–658, 2006), the generalized median estimator (Brazauskas and Serfling in Extremes 3:231–249, 2001), the partial density component estimator (Vandewalle et al. in Comput Stat Data Anal 51:6252–6268, 2007), and the probability integral transform statistic estimator (PITSE) (Finkelstein et al. in N Am Actuar J 10:1–10, 2006). Monte Carlo simulations show that the PITSE offers the desired compromise between ease of use and power to protect against outliers in the small-sample setting.
Highlights
Distributions of many economic variables are characterized by heavy right tails
For higher robustness and lower asymptotic relative efficiency (ARE) (Table 4), when the weighted maximum likelihood estimator (WMLE) is included in the comparison, we can observe that the WMLE performs worse than the alternatives, especially in terms of RRMSE
The classical Pareto distribution is widely used in many areas of economics and other sciences to model the right tail of heavy-tailed distributions
Summary
Distributions of many economic variables are characterized by heavy right tails. Such tails are often modelled in economics and other fields of science using Pareto distribution, which was originally introduced late in the nineteenth century by Vilfredo Pareto in the context of modelling income and wealth distributions (Pareto 1897). As observed recently by Beran and Schell (2012), researchers and practitioners studying problems such as operational risk assessment, reinsurance and natural disasters often have to fit heavy-tailed models to sparse samples with the number of observations ranging from 20 to at most 50 In another context, Barro and Jin (2011) have estimated the upper-tail exponent of the distribution of macroeconomic disasters using samples of only 21–22 observations. A recent study of Ogwang’s (2011), which analyses the Pareto behaviour of the top Canadian wealth distribution is based on a rather small sample of about one hundred observations It seems that in practical applications the Pareto tail index is quite often estimated using sparse data. The present paper fills the gap in the literature by providing an extensive comparison of the small-sample properties of the most popular robust estimators for the Pareto tail index. Distribution data from the European Union Statistics on Income and Living Conditions (EU-SILC), while Sect. 5 concludes and gives recommendations for practice
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