Efficient and robust estimation of tail parameters for Pareto and exponential models

In this paper, a new efficient and robust estimator of the Pareto tail index is proposed. Although the emphasis is on the Pareto distribution, all results are valid for the estimation of the scale/rate parameter of the two-parameter exponential distribution. The approach is to assume that the observations were generated from the FLLP-contaminated Pareto, that is, a mixture of the Pareto and FLLP distributions. The latter is an original distribution designed specifically to represent any outlier distribution. The parameters are estimated using an iterative process adapted from the expectation-maximization (EM) algorithm to optimize the properties of the estimators in a robustness context. A robust confidence interval for the Pareto tail index is also given. It is shown through different asymptotic results that these estimators reach a breakdown point of 50% with full efficiency. Their simultaneous high efficiency and high robustness are also shown for finite samples in a large Monte Carlo simulation study. Finally, an example with a real dataset of daily crude oil returns is given.