  In this paper, we propose an estimator of the second-order parameter of
randomly right-truncated Pareto-type distributions data and establish its
consistency and asymptotic normality. Moreover, 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 the 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.
