The application of nonparametric estimates of the probability density function requires the evaluation of smoothing parameters like bandwidths of kernel estimates. We consider the so-called discrepancy method. The latter was proposed and investigated in
Vapnik, Markovich as an alternative data-dependent smoothing tool to cross-validation. It is based on the usage of well-known nonparametric statistics like the von Mises-Smirnov's (M-S) and the Kolmogorov-Smirnov (K-S) as measures in the space of distribution functions. The unknown smoothing parameter is proposed to find as a solution of the discrepancy equation. On its left-hand side it stands a discrepancy between the empirical distribution function and the nonparametric estimate of the distribution function. The latter is obtained as a corresponding integral of the density nonparametric estimator. The right-hand side is equal to some quantile of the asymptotic distribution of the M-S or K-S statistic.
The discrepancy method demonstrates better results than cross-validation for nonsmooth (e.g., triangular and uniform) distributions. The method can avoid the problem of cross-validation falling into local extremes. In Vapnik it is derived that the rate of convergence in $L_2$ for a projection estimator with the smoothing parameter found by the M-S discrepancy method is close to the best in the class of densities with a bounded variation of the kth derivative.
The properties of the discrepancy method in case of heavy-tailed densities is also discussed. The application of the discrepancy method to the nonparametric estimation of the extremal index is proposed.
