An inversion theorem-based kernel density estimator for a weighted average and difference of weighted averages with applications

In this article, we develop a new and novel kernel density estimator for a sum of weighted averages from a single population based on utilizing the well defined kernel density estimator in conjunction with classic inversion theory. This idea is further developed for a kernel density estimator for the difference of weighed averages from two independent populations. The resulting estimator is bootstrap-like in terms of its properties with respect to the derivation of approximate confidence intervals via a "plug-in" approach. This new approach is distinct from the bootstrap methodology in that it is analytically and computationally feasible to provide an exact estimate of the distribution function through direct calculation. Thus, our approach eliminates the error due to Monte Carlo resampling that arises within the context of simulation based approaches that are oftentimes necessary in order to derive bootstrap-based confidence intervals for statistics involving weighted averages of i.i.d. random variables. We provide several examples and carry forth a simulation study to show that our kernel density estimator performs better than the standard central limit theorem based approximation in term of coverage probability.