Kernel Density Estimation with Linked Boundary Conditions
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Kernel density estimation on a finite interval poses an outstanding challenge because of the well-recognized boundary bias issues at the end-points. Motivated by an application of density estimation in biology, we consider a new type of boundary constraint, in which the values of the estimator at the two boundary points are linked. We provide a kernel density estimator that successfully incorporates this linked boundary condition. This is studied via a nonsymmetric heat kernel which generates a series expansion in nonseparable generalised eigenfunctions of the spatial derivative operator. Despite the increased technical challenges, the model is proven to outperform the more familiar Gaussian kernel estimator, yet it inherits many desirable analytical properties of the latter KDE model. We apply this to our motivating example in biology, as well as providing numerical experiments with synthetic data. The method is fast and easy to use and we also compare against existing methods, which do not incorporate this constraint and are therefore inaccurate near the boundary.
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