Estimating the second-order parameter in statistics of extremes under random truncation and application

In this paper, we propose an estimator of the second-order parameter of Pareto-type distributions for randomly right-truncated data and establish their consistency and asymptotic normality. As application, we derive an asymptotically unbiased estimator of the tail index and study its asymptotic behaviour. Our considerations are based on a useful Gaussian approximation of a tail product-limit process recently given by Benchaira et al. [Tail product-limit process for truncated data with application to extreme value index estimation. Extremes, 2016; 19: 219-251] and the results of Gomes et al. [Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes, 2002; 5: 387-414]. We show, by simulation, that the proposed estimators behave well, in terms of bias and mean square error.