On Estimating L_2^2 Divergence

Akshay Krishnamurthy, Kirthevasan Kandasamy, Barnabas Poczos, Larry Wasserman
Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, PMLR 38:498-506, 2015.

Abstract

We give a comprehensive theoretical characterization of a nonparametric estimator for the L_2^2 divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is \sqrtn-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Esséen style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal.

Cite this Paper


BibTeX
@InProceedings{pmlr-v38-krishnamurthy15, title = {{On Estimating L_2^2 Divergence}}, author = {Akshay Krishnamurthy and Kirthevasan Kandasamy and Barnabas Poczos and Larry Wasserman}, booktitle = {Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics}, pages = {498--506}, year = {2015}, editor = {Guy Lebanon and S. V. N. Vishwanathan}, volume = {38}, series = {Proceedings of Machine Learning Research}, address = {San Diego, California, USA}, month = {09--12 May}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v38/krishnamurthy15.pdf}, url = { http://proceedings.mlr.press/v38/krishnamurthy15.html }, abstract = {We give a comprehensive theoretical characterization of a nonparametric estimator for the L_2^2 divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is \sqrtn-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Esséen style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal.} }
Endnote
%0 Conference Paper %T On Estimating L_2^2 Divergence %A Akshay Krishnamurthy %A Kirthevasan Kandasamy %A Barnabas Poczos %A Larry Wasserman %B Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2015 %E Guy Lebanon %E S. V. N. Vishwanathan %F pmlr-v38-krishnamurthy15 %I PMLR %P 498--506 %U http://proceedings.mlr.press/v38/krishnamurthy15.html %V 38 %X We give a comprehensive theoretical characterization of a nonparametric estimator for the L_2^2 divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is \sqrtn-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Esséen style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal.
RIS
TY - CPAPER TI - On Estimating L_2^2 Divergence AU - Akshay Krishnamurthy AU - Kirthevasan Kandasamy AU - Barnabas Poczos AU - Larry Wasserman BT - Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics DA - 2015/02/21 ED - Guy Lebanon ED - S. V. N. Vishwanathan ID - pmlr-v38-krishnamurthy15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 38 SP - 498 EP - 506 L1 - http://proceedings.mlr.press/v38/krishnamurthy15.pdf UR - http://proceedings.mlr.press/v38/krishnamurthy15.html AB - We give a comprehensive theoretical characterization of a nonparametric estimator for the L_2^2 divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is \sqrtn-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Esséen style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal. ER -
APA
Krishnamurthy, A., Kandasamy, K., Poczos, B. & Wasserman, L.. (2015). On Estimating L_2^2 Divergence. Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 38:498-506 Available from http://proceedings.mlr.press/v38/krishnamurthy15.html .

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