We present general sufficient conditions for the almost sure $L_1$-consistency of histogram density estimates based on data-dependent partitions. Analogous conditions guarantee the almost-sure risk consistency of histogram classification schemes based on data-dependent partitions. Multivariate data are considered throughout.
In each case, the desired consistency requires shrinking cells, subexponential growth of a combinatorial complexity measure and sublinear growth of the number of cells. It is not required that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points. No assumptions are made concerning the common distribution of the training vectors.
We apply the results to establish the consistency of several known partitioning estimates, including the $k_n$-spacing density estimate, classifiers based on statistically equivalent blocks and classifiers based on multivariate clustering schemes.
"Consistency of data-driven histogram methods for density estimation and classification." Ann. Statist. 24 (2) 687 - 706, April 1996. https://doi.org/10.1214/aos/1032894460