The study of distribution tails is a fundamental research in statistical frequency analysis relevant to many research fields, such as insurance, hydrological events, earthquake, etc. Here, we describe and investigate the effect and feasibility of the high-order L-moment (LH-moment) method for estimating heavy-tail conditions by fitting a four parameter kappa distribution. Details of parameter estimation using LH-moments for the four parameter kappa distribution (K4D) are described and formulated. Monte-Carlo simulation is performed to illustrate the performance of the LH-moment method in terms of heavy-tail quantiles over all quantiles using K4D and non K4D samples, respectively. The result suggests that the method is either useful (when the method of L-moment estimation fails to give a feasible solution) or as effective as the L-moment approach in handling data following K4D. Applications to the annual maximum flood and sea level data are presented.