Minimum local distance density estimation

ABSTRACT

We present a local density estimator based on first-order statistics. To estimate the density at a point, x, the original sample is divided into subsets and the average minimum sample distance to x over all such subsets is used to define the density estimate at x. The tuning parameter is thus the number of subsets instead of the typical bandwidth of kernel or histogram-based density estimators. The proposed method is similar to nearest-neighbor density estimators but it provides smoother estimates. We derive the asymptotic distribution of this minimum sample distance statistic to study globally optimal values for the number and size of the subsets. Simulations are used to illustrate and compare the convergence properties of the estimator. The results show that the method provides good estimates of a wide variety of densities without changes of the tuning parameter, and that it offers competitive convergence performance.

Keywords:

• Density estimation
• nearest-neighbor
• order statistics.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G07
• 62G30

Acknowledgments

We thank Y. Marzouk for reviewing our proofs, and D. Allaire, L. Ng, C. Lieberman and R. Stogner for helpful discussions. The first and third authors acknowledge the support of the the DOE Applied Mathematics Program, Awards DE-FG02-08ER2585 and DE-SC0009297, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center.