The Essential Histogram

Abstract

The histogram is widely used as a simple, exploratory display of data, but it is usually not clear how to choose the number and size of bins for this purpose. We construct a confidence set of distribution functions that optimally address the two main tasks of the histogram: estimating probabilities and detecting features such as increases and (anti)modes in the distribution. We define the essential histogram as the histogram in the confidence set with the fewest bins. Thus the essential histogram is the simplest visualization of the data that optimally achieves the main tasks of the histogram. We provide a fast algorithm for computing a slightly relaxed version of the essential histogram, which still possesses most of its beneficial theoretical properties, and we illustrate our methodology with examples. An R-package is available online.