<p>For random processes where extreme events are possible, heavy-tailed distributions are common. Without loss of generality, when the interest is the behavior of the right tail of a heavy-tailed distribution, the desire is usually the estimation of a certain threshold beyond which the extreme events are observed, and the attendant action is usually to model these extreme observations with a generalized Pareto distribution. While some graphical methods, such as the mean excess plot, the stability plot, and the Hill plot, etc., exist for the estimation of this threshold, it remains desirable for this threshold to be estimated algorithmically as part of the parameters of a hybrid model which accounts for both the main behavior of the data and the tail behavior. Thus, after proper consideration of the dissymmetry of the data, a two-component hybrid model can be applied for modeling the entire distribution of the data. In this paper, a new two-component hybrid model that links the half-Cauchy distribution (for the main behavior) and the generalized Pareto distribution (for the tail behavior) is proposed and used to model heavy-tailed distributions. An unsupervised iterative estimation algorithm based on the Levenberg-Marquardt framework is adopted for the estimation of the parameters of the hybrid model, and in addition, the parameters to be estimated include the threshold. Results from Monte Carlo simulations are obtained, and the application to stocks and insurance claims data sets is carried out. For both data sets, results from the proposed hybrid model are compared with those from a two-component version employing a half-normal rather than a half-Cauchy distribution for the main behavior of the data. Graphical threshold estimation methods from extreme value theory are likewise applied and benchmarked against the model-based threshold estimate using the estimation algorithm.</p>