Nearest neighbor density functional estimation from inverse Laplace transform
This provides a unified framework for density functional estimation, which is incremental but extends to new functionals like logarithmic entropies and divergences, benefiting statistical and machine learning applications.
The paper tackles the problem of estimating general density functionals using k-nearest neighbor distances, proposing a new asymptotically unbiased estimator based on inverse Laplace transforms. It recovers existing estimators for entropies and divergences, discovers new ones, and establishes L2-consistency and convergence rates for smooth, bounded densities.
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 Rényi entropies and Kullback--Leibler and Rényi 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.