Kernel density estimation with linked boundary conditions

Abstract

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 nonseparable 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-theart 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 a priori knowledge. Our analysis can also be extended to more general This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line

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“…The application of the UTM is systematic, regardless of the types of boundary data, e.g., nonhomogeneous Dirichlet, Neumann, and Robin conditions. This is one reason the UTM is more general and effective than standard methods for evolution IBVPs 18 . Further, the method demonstrates how many and which types of boundary conditions result in a well‐posed IBVP, depending on the order of the PDE 14 …”

“…This is one reason the UTM is more general and effective than standard methods for evolution IBVPs. 18 Further, the method demonstrates how many and which types of boundary conditions result in a well-posed IBVP, depending on the order 𝑁 of the PDE. 14 For either half-line or finite-interval IBVPs, the UTM is applied algorithmically using the following steps 19 :…”

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