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Nearest neighbor density functional estimation from inverse Laplace transform
- Publication Year :
- 2018
-
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.<br />Comment: 43 pages, 4 figures. IEEE Transactions on Information Theory (to appear)
Details
- Database :
- OAIster
- Publication Type :
- Electronic Resource
- Accession number :
- edsoai.on1106299421
- Document Type :
- Electronic Resource
- Full Text :
- https://doi.org/10.1109.TIT.2022.3151231