Locating bright spots in a point process

Abstract

As an approach to modelling the ‘matching’ of optical receptors in animals to the objects they are designed to see, we study the problem of locating regions of high intensity in a point process on the real line, using the counts of points in a movable interval of fixed length. We define performance measures analogous to statistical size and power for this procedure and, for points forming a renewal process, give conditions on the quantiles of the convolutions of the interpoint distribution which ensure that the optimal length for the ‘detector’ is close to that of the ‘object’ to be detected. We show that these conditions are satisfied for a Poisson process. Similar conditions ensure that the optimal length is close to zero, and we give a class of distributions satisfying these conditions. Finally we show that the results can be extended to simple two-dimensional models.