Nearest neighbor density functional estimation from inverse Laplace transform

Code

• github.com/jongharyu/knn-functional-estimation
  OfficialIn papernone★ 0

Abstract

A new approach to L_2-consistent estimation of a general density functional using k-nearest neighbor distances is proposed, where the functional under consideration is in the form of the expectation of some function f of the densities at each point. The estimator is designed to be asymptotically unbiased, using the convergence of the normalized volume of a k-nearest neighbor ball to a Gamma distribution in the large-sample limit, and naturally involves the inverse Laplace transform of a scaled version of the function f. Some instantiations of the proposed estimator recover existing k-nearest neighbor based estimators of Shannon and R\'enyi entropies and Kullback--Leibler and R\'enyi divergences, and discover new consistent estimators for many other functionals such as logarithmic entropies and divergences. The L_2-consistency of the proposed estimator is established for a broad class of densities for general functionals, and the convergence rate in mean squared error is established as a function of the sample size for smooth, bounded densities.

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