5 research outputs found
Kernel Density Estimation with Linked Boundary Conditions
Kernel density estimation on a finite interval poses an outstanding challenge
because of the well-recognized bias at the boundaries of the interval.
Motivated by an application in cancer research, we consider a boundary
constraint linking the values of the unknown target density function at the
boundaries. We provide a kernel density estimator (KDE) that successfully
incorporates this linked boundary condition, leading to a non-self-adjoint
diffusion process and expansions in non-separable generalized eigenfunctions.
The solution is rigorously analyzed through an integral representation given by
the unified transform (or Fokas method). The new KDE possesses many desirable
properties, such as consistency, asymptotically negligible bias at the
boundaries, and an increased rate of approximation, as measured by the AMISE.
We apply our method to the motivating example in biology and provide numerical
experiments with synthetic data, including comparisons with state-of-the-art
KDEs (which currently cannot handle linked boundary constraints). Results
suggest that the new method is fast and accurate. Furthermore, we demonstrate
how to build statistical estimators of the boundary conditions satisfied by the
target function without apriori knowledge. Our analysis can also be extended to
more general boundary conditions that may be encountered in applications