Asymptotically Optimal Cells for a Historgram

Abstract

The purpose of this paper is to examine the properties of the histogram when the cells are allowed to be arbitrary. Given a random sample from an unknown probability density $f$ on $I$, we wish to construct a histogram. Any partition of $I$ can be used as cells. The optimal partition minimizes the mean integrated squared error (MISE) of the histogram from $f$. An expression is found for the infimum of MISE over all partitions. It is proved that the infimum is attained asymptotically by minimizing MISE over a class of partitions of locally equisized cells.