Weak Convergence of the Empirical Mean Excess Process with Application to Estimate the Negative Tail Index

Let Y _i , 1 ≤ i ≤ n be i.i.d. random variables with the generalized Pareto distribution W _γ,σ with γ < 0. We define the empirical mean excess process with respect to {Y _i , 1 ≤ i ≤ n} as in Eq. 2.1 (see below) and investigate its weak convergence. As an application, two new estimators of the negative tail index γ are constructed based on the linear regression to the empirical mean excess function and their consistency and asymptotic normality are obtained.