To estimate a heavy-tailed probability density function (pdf), different approaches are summarized: (1) a combined parametric - nonparametric method, (2) methods based on data transformations and, (3) a variable bandwidth kernel estimator.
The first method implies a separate estimation of the 'tail' and 'body' of the pdf by parametric and nonparametric methods, respectively. We consider a Pareto-type model to fit the 'tail' and a finite series expansion in terms of trigonometric functions as 'body' estimate. To fit the body of a multi-modal pdf better, we use a structural risk minimization method for the selection of the parameters.
The second approach requires a special data transformation which improves the estimation in the 'tails', namely, the transformation from a Generalized Pareto distribution function (df) which is assumed as a fitted df to a triangular df selected as the target df. The latter transformation is robust regarding the uncertainty of the tail index estimation. The triangular pdf can be estimated by a nonparametric estimator, e.g., a Parzen kernel estimator or a polygram. Regarding the heavy-tailed pdf estimation a kernel estimator with
a variable bandwidth is usually recommended due to the variability of its bandwidth for each observation. It is demonstrated that this estimator works better if a preliminary data transformation is used.
To select data-driven smoothing parameters for the mentioned estimators, a discrepancy method is considered as an alternative to the cross-validation method. The discrepancy method is based on nonparametric statistics like the Kolmogorov-Smirnov or the von Mises-Smirnov statistics, and it uses quantiles of their limit distributions as a unknown discrepancy between the fitted and empirical dfs. Moreover, the convergence rates of these estimates are discussed.
