Using the scan statistic to test for uniformity
The problem of "bump hunting", i.e., finding a rectangular signal in the presence of background noise, is an important activity of many experimental physicists. Suppose it is desired to test whether points in an interval are uniformly distributed, versus whether they come from a clustering (or bump) alternative. Take a fixed scan width and slide a window across the points, noting how many points are in the window as a function of its location. Then the scan statistic is the maximum, over window location, of the number of points in the window. To test the uniformity hypothesis against the clustering alternative, it can be shown that the generalized likelihood ratio test yields the scan statistic. This article discusses properties of the statistic, and gives directions for further research.