We consider density estimators based on the nearest neighbors method appliedto discrete point distibutions in spaces of arbitrary dimensionality. If thedensity is constant, the volume of a hypersphere centered at a random locationis proportional to the expected number of points falling within the hypersphereradius. The distance to the $N$-th nearest neighbor alone is then a sufficientstatistic for the density. In the non-uniform case the proportionality isdistorted. We model this distortion by normalizing hypersphere volumes to thelargest one and expressing the resulting distribution in terms of the Legendrepolynomials. Using Monte Carlo simulations we show that this approach can beused to effectively address the tradeoff between smoothing bias and estimatorvariance for sparsely sampled distributions.
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