Open Access
March 2024 Efficient and robust estimation of tail parameters for Pareto and exponential models
Alain Desgagné
Author Affiliations +
Braz. J. Probab. Stat. 38(1): 148-176 (March 2024). DOI: 10.1214/24-BJPS597


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.

Funding Statement

The author was supported by the Canadian Institute of Actuaries.


The financial support of the Canadian Institute of Actuaries is gratefully acknowledged. The author would also like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.


Download Citation

Alain Desgagné. "Efficient and robust estimation of tail parameters for Pareto and exponential models." Braz. J. Probab. Stat. 38 (1) 148 - 176, March 2024.


Received: 1 October 2022; Accepted: 1 January 2024; Published: March 2024
First available in Project Euclid: 4 March 2024

MathSciNet: MR4718430
Digital Object Identifier: 10.1214/24-BJPS597

Keywords: heavy tails , M-estimator , Monte Carlo simulations , Outliers , relative excesses over a large threshold , robust weighted maximum likelihood estimator

Rights: Copyright © 2024 Brazilian Statistical Association

Vol.38 • No. 1 • March 2024
Back to Top