Estimation of Tails and Related Quantities Using the Number of Near-Extremes

Abstract

In an insurance context, consider {X_n, n ≥ 1} random claim sizes with common distribution function F and {N(t), t ≥ 0} an integer valued stochastic process tfiat counts the number of claims occurring during the time interval [0, t]. Based on the number of near-extremes which are the observations X_i near the largest or the mth largest observation, we derive in this article a strongly consistent estimator of upper tails of X₁. Furthermore, estimators for both the tail index and the upper endpoint are introduced when F is a generalized Pareto distribution. Asymptotic normal law for the proposed estimators is additionally presented.

Key Words:

• Asymptotic normality
• Estimation of tail index
• Estimation of the upper endpoint
• Generalized Pareto distribution
• Number of near-extremes