Achieving Higher-Order Convergence Rates for Density Estimation with Binned Data

Abstract

A method is proposed for achieving fast— O ( n –8/9 ) and faster—asymptotic mean integrated squared error (AMISE) convergence rates in density estimation using data condensed to standard histogram bin counts and edges. The method involves the two elements of B splines and the correct mass proportions property, in which the probability mass for the histogram bins equals exactly the fraction of the data found in those bins. It is shown that the asymptotic bias of such an estimate using a spline of order p and bin width h will be O ( h P ), resulting in optimal AMISE of O ( n –2 p /2 p +1 ) form. Theoretical and numeric comparisons to kernel-based and other estimators show that the histospline estimator is usually comparable and often superior. An extension to the multivariate density estimation setting is described.