The Regular Histogram Rule

Abstract

In this chapter we study the cubic histogram rule. Recall that this rule partitions R ^d into cubes of the same size, and gives the decision according to the number of zeros and ones among the Y _i ’s such that the corresponding X _i falls in the same cube as X. P _n = {A _n1, A _n2,...} denotes a partition of R ^d into cubes of size h _n > 0, that is, into sets of the type $$, where the k _i ’s are integers, and the histogram rule is defined by

$$$$

where for every x ∈ R ^d, A _n (x) ∈ A _ni if x ∈ A _ni . That is, the decision is zero if the number of ones does not exceed the number of zeros in the cell in which x falls. Weak universal consistency of this rule was shown in Chapter 6 under the conditions h _n → 0 and nh ^d _n → ∞ as n → ∞. The purpose of this chapter is to introduce some techniques by proving strong universal consistency of this rule. These techniques will prove very useful in handling other problems as well. First we introduce the method of bounded differences.