Asymptotic and qualitative performance of non-parametric density estimators: a comparative study

Motivated by finance applications, we assessed the performance of several univariate density estimation methods, focusing on their ability to deal with heavy-tailed target densities. Four approaches, a fixed bandwidth kernel estimator, an adaptive bandwidth kernel estimator, the Hermite series (SNP) estimator of Gallant and Nychka, and the logspline estimator of Kooperberg and Stone, are compared. We conclude that the logspline and adaptive kernel methods provide superior performance, and the convergence rate of the SNP estimator is remarkably slow compared with the other methods. The Hellinger convergence rate of the SNP estimator is derived as a function of tail heaviness. These findings are confirmed in Monte Carlo experiments. Qualitative assessment reveals the possibility that side lobes in the tails of the fixed kernel and SNP estimates are artefacts of the fitting method.