247 – Toward Better Communication of Information with Statistical Graphics
What Can We Learn from Correct Calculation of Histograms?
James Weber
University of Illinois Chicago, University of Wisconsin Milwaukee
Defining "good" histograms often has focused on asymptotic results regarding bin width, as well as integrated measures of error, such as MISE. MISE leads to a statistical objective function to optimize with data to obtain a "best" histogram density and equivalent frequency histogram. Bin edge discontinuity significantly interferes with achieving good approximations. However exact MISE and other kinds of good histograms can be obtained so that explicit error in determining estimated MISE histograms for small to moderate samples size can be considered. That is, uniform bin width histograms are so easy to construct that apparently little thought has been given to exact calculation of, for example, MISE error minimizing histograms. Exact calculations show that reasonable approximations are so far away from exact (from the perspective of only shape) that exact should be used for any histogram wanted to satisfy statistical criteria such as MISE, maximum likelihood, method of moments, shape stability, min chi-squared, other least squares, etc.