Bootstrap Confidence Regions for the Intensity of a Poisson Point Process

We develop bootstrap methods for constructing conndence regions, including intervals and simultaneous bands, in the context of estimating the intensity function of a non-stationary Poisson process. Several diierent 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 much the same way one would use the bootstrap in problems involving independent and identically distributed observations. For each diierent bootstrap method a variety of percentile-t ways of constructing conndence bands is described, producing bands whose width varies in proportion to standard deviation, or is approximately constant, depending on the application. The eeectiveness of these diierent approaches is demonstrated both theoretically and numerically, for real and simulated data. Issues such as bias correction are addressed.