Robust kernel estimator for densities of unknown smoothness

Abstract

1Results on nonparametric kernel estimators of density differ according to the assumed degree of density smoothness; it is often assumed that the density function is at least twice differentiable. However, there are cases where non-smooth density functions may be of interest. We provide asymptotic results for kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair. We also derive the limit joint distribution of kernel density estimators corresponding to different bandwidths and kernel functions. Using these results, we construct an estimator that combines several estimators for different bandwidth/kernel pairs to protect against the negative consequences of errors in assumptions about order of smoothness. The results of a Monte Carlo experiment confirm the usefulness of the combined estimator. We demonstrate that while in the standard normal case the combined estimator has a relatively higher mean squared error than the standard kernel estimator, both estimators are highly accurate. On the other hand, for a non-smooth density where the MSE gets very large, the combined estimator provides uniformly better results than the standard estimator.

Rates of Expansions for Functional Estimators

“…This theoretical result shows the possibility of improved performance of averaged estimators, also referred to as combined estimators. Kotlyarova and Zinde-Walsh (2007) and Schafgans and Zinde-Walsh (2010) construct robust estimators of densities and density weighted average derivatives using estimated weights that minimize the asymptotic MSE (AMSE), and Kotlyarova et al (2016) generalize the results to local and general averaged kernel based estimators.…”

“…Marron and Wand (1992) and Härdle et al (1998) in the statistics literature investigated the estimation of density for cases with large derivatives, but such distributions are rarely considered elsewhere. This has prompted Kotlyarova and Zinde-Walsh (2007), Schafgans and Zinde-Walsh (2010), and Kotlyarova et al (2016) to explore performance of kernel estimators and kernel-based statistics in cases of normal mixture with peaked normals where derivatives could vastly exceed those for the standard cases. Simulations with such mixtures produce dramatically different results from those in the literature.…”

“…Parzen (1962), Rosenblatt (1956), Kotlyarova and Zinde-Walsh (2007) (non-smooth), Zinde-Walsh (2008, 2017) (gen fun)…”

Smoothness adaptive average derivative estimation

“…We consider different kernels as well as different bandwidths in linear combinations, since the selection among kernels (higher and lower order) is also hampered by an unknown degree of smoothness. This is an important generalization, in particular given that the order of the kernel has been shown to have a large impact on the finite sample performance for density estimation and similarly, for kernels of the same order, different shapes (including asymmetric) affect performance; see Hansen (2005) and Kotlyarova and Zinde-Walsh (2007).…”

“…With moderate sample size we use two low‐order kernels. For large samples a variety of kernels including asymmetric kernels could be beneficial—the order and shapes affect performance; see Hansen (2005) and Kotlyarova and Zinde‐Walsh (2007). Since minimizing means in effect minimizing a ′ Da of , which has exactly the same structure as in KZW, their theorem 3.2 applies to show that the optimal weights provide the best convergence rate for a ′ Da available for any included bandwidth.…”