Bootstrap confidence regions for the intensity of a Poisson point process

Bootstrap methods are developed for constructing confidence regions for the intensity function of a nonstationary Poisson process. Several different resampling algorithms are suggested, ranging from resampling a Poisson process with intensity equal to that estimated nonparametrically from the data to resampling the data points themselves in the same manner that the bootstrap is used in problems involving independent and identically distributed observations. For each different bootstrap method, various percentile-t ways of constructing confidence bands are described. The effectiveness of these different approaches is demonstrated both theoretically and numerically, for real and simulated data. Issues such as bias correction are addressed.