Asymptotically Optimal Balloon Density Estimates

Given a sample of n observations from a density ƒ on Rd, a natural estimator of ƒ(x) is formed by counting the number of points in some region R surrounding x and dividing this count by the d dimensional volume of R. This paper presents an asymptotically optimal choice for R. The optimal shape turns out to be an ellipsoid, with shape depending on x. An extension of the idea that uses a kernel function to put greater weight on points nearer x is given. Among nonnegative kernels, the familiar Bartlett-Epanechnikov kernel used with an ellipsoidal region is optimal. When using higher order kernels, the optimal region shapes are related to Lp balls for even positive integers p.