Nonparametric estimation of functionals of densities from finite number of samples is an important tool in domains such as statistics, signal processing and machine learning. While several estimators have been proposed in literature, the performance of these estimators is not known. We propose a simple kNN density estimation based plug-in estimator for estimation of non-linear functionals of densities. Based on the properties of kNN density estimates, we derive the bias, variance and mean square error of the estimator in terms of the sample size, the dimension of the samples and the underlying probability distribution. Based on these results, we specify the optimal choice of tuning parameters for minimum mean square error. We also present results on convergence in distribution of the proposed estimator. Rates of convergence to the asymptotic distribution are obtained.
Kumar Sricharan, Raviv Raich, and Alfred O. Hero III, “Global performance prediction for divergence-based image registration criteria,” in IEEE Workshop on Statistical Signal Processing (SSAP), Cardiff, UK. Sept. 2009. (.pdf)
Figure 1. Variation of bias of estimated MI vs M for fixed N = 1000 with ±95% confidence envelopes.
Figure 2. Variation of variance of estimated MI vs N for fixed M = 10000 and bandwidth k = 411 with ±95% confidence envelopes.
Figure 3. Q-Q plot of normalized MI estimate and standard normal distribution.
Figure 4. Variation of MI with mixing ratio p with ±95% confidence envelopes.