On Detecting Periodicity in Astronomical Point Processes
We consider the problem of detecting periodicity in the rate function of a point process or a marked point process, motivated by the problem of detecting $\gamma$-ray pulsars. The detection problem poses both theoretical and computational challenges. On the theoretical side, there are no compelling optimality results that dictate the choice of a detection algorithm and the properties of detection procedures can be quite difficult to analyze. On the computational side, searching over a range of frequency and frequency drift can be a daunting task, even for a record consisting of only a thousand or so events. We discuss a class of detection procedures, weighted quadratic test statistics arising from likelihood expressions, whose properties we can understand and which do not impose excessive computational burdens. We show how knowledge of the point spread function associated with photon arrivals can be incorporated to improve power. We show that if a search over frequencies is conducted by discretizing a frequency band, the discretization must be very fine and we discuss the use of integration over frequency bands as an alternative. We also discuss the use of extreme value theory in conjunction with simulation in assessing statistical significance for such a search.