Goodness-of-fit for Sparse Distributions in High Energy Physics

We consider Pearson’s chi-square X2, the likelihood ratio G2, and Zelterman’s D2 as goodness-of-fit statistics for high energy physics problems in several dimensions, where the data are sparse. There is a fundamental obstacle in the “ultrasparse” case where all bins have at most one entry (ni = 0, 1 ∀i). A condition for avoiding this regime is derived; the allowed number of bins k rises faster than the total number of events n: kmax = 0.4× n1.4. Reasonable binning in many dimensions may thus be possible for modest datasets n > O(100), although special treatment is required to derive p-values. Results for an initial trial problem are encouraging; further studies are underway.