  A Local Orthogonal Polynomial Expansion (LOrPE) of the empirical density
function is proposed as a novel method to estimate the underlying density. The
estimate is constructed by matching localized expectation values of orthogonal
polynomials to the values observed in the sample. LOrPE is related to several
existing methods, and generalizes straightforwardly to multivariate settings.
By manner of construction, it is similar to Local Likelihood Density Estimation
(LLDE). In the limit of small bandwidths, LOrPE functions as Kernel Density
Estimation (KDE) with high-order (effective) kernels inherently free of
boundary bias, a natural consequence of kernel reshaping to accommodate
endpoints. Faster asymptotic convergence rates follow. In the limit of large
bandwidths, LOrPE is equivalent to Orthogonal Series Density Estimation (OSDE)
with Legendre polynomials. We compare the performance of LOrPE to KDE, LLDE,
and OSDE, in a number of simulation studies. In terms of mean integrated
squared error, the results suggest that with a proper balance of the two tuning
parameters, bandwidth and degree, LOrPE generally outperforms these competitors
when estimating densities with sharply truncated supports.
