Nonparametric Estimation of Heavy-Tailed Density by the Discrepancy Method

The nonparametric estimation of the probability density function (pdf) requires smoothing parameters like bandwidths of kernel estimates. We consider the so-called discrepancy method proposed in [13, 14, 21] as a data-driven smoothing tool and alternative to cross-validation. It is based on the von Mises-Smirnov's (M-S) and the Kolmogorov-Smirnov's (K-S) nonparametric statistics as measures in the space of distribution functions (cdfs). The unknown smoothing parameter is found as a solution of the discrepancy equation. On its left-hand side stands the measure between the empirical distribution function and the nonparametric estimate of the cdf. The latter is obtained as a corresponding integral of the pdf estimator. The right-hand side is equal to a quantile of the asymptotic distribution of the M-S or K-S statistic. The discrepancy method considered earlier for light-tailed pdfs is investigated now for heavy-tailed pdfs.