Finite-sample performance of the T- and W-estimators for the Pareto tail index under data truncation and censoring

Abstract

A single-parameter Pareto model, Pareto I, arises in many areas of application such as pricing of insurance risks, measuring income or wealth inequality in economics, or modelling lengths of telephone calls in telecommunications. In insurance, for example, it is common to work with data that are truncated (due to deductibles), censored (due to policy limits), and contaminated by outliers (due to model misspecification). Therefore, it is prudent to estimate Pareto I using robust procedures that are designed for such data transformations and are resistant to outliers. In this paper, we consider trimmed (T) and winsorized (W) estimators for the Pareto tail index α, and conduct a simulation study to check these estimators' performance in finite samples. Two broad aspects are investigated:

1. Convergence of $\hat{\alpha }$'s to α as sample size increases from n = 25 to n = 1000. This is evaluated by presenting a series of boxplots and measuring each estimator's relative bias and finite-sample relative efficiency (with respect to the asymptotic variance of the maximum likelihood estimator, MLE).

2. Sensitivity of $\hat{\alpha }$'s and of two value-at-risk estimates to a new observation (an outlier) that is placed at various locations within the sample ranging from the point of left-truncation/-censoring to the point of right-censoring.

We find that with respect to (a), T- and W-estimators perform best when n is small and MLE outperforms its competitors when n is large. And with respect to (b), MLE is not robust whilst T's and W's can offer any desired degree of robustness.

Keywords:

• Insurance payments
• Pareto tail index
• robust estimation
• trimmed and winsorized moments
• truncated and censored data

Acknowledgments

The authors are very appreciative of valuable insights and useful comments provided by the anonymous referee, which helped to substantially improve the paper. Also, much of this work was completed while the first author was a Ph.D. student in the Department of Mathematical Sciences at the University of Wisconsin-Milwaukee. 

Disclosure statement

No potential conflict of interest was reported by the author(s).