Bootstrap Confidence Regions for the Intensity of a Poisson Point Process

Article excerpt

1. INTRODUCTION

Nonparametric methods for estimating a varying point process intensity have been developed by a number of authors, including Diggle (1985), Diggle and Marron (1988), Ellis (1986), and Leadbetter and Wold (1983). Point estimates may be difficult to interpret without some idea of their accuracy. That typically requires confidence intervals or bands, and the problem of constructing such regions can be rather awkward in the context of dependent data, such as that derived from a Poisson process. Here we develop several simple and, we argue, attractive bootstrap methods for computing confidence regions for Poisson intensity functions. They use a variety of different resampling algorithms and different variants of the percentile-t bootstrap.

Thus the context of our article is that of the bootstrap for dependent processes. Although we treat point processes rather than time series, the setting is in principle not unlike that of bootstrapping a time series when a structural model is available for the type of dependence (e.g., an autoregression), yet the distribution of perturbations is virtually arbitrary. This type of problem has been treated by, for example, Bose (1988), but there are fundamental differences between that work and our own. In particular, in the point process context, with no structural assumptions made about the intensity function, the quantity of interest can be accessed only via statistical smoothing, and so smoothing is an essential feature of all our algorithms. This is not the case with more traditional inferential problems involving dependent data. There are of course other bootstrap approaches to statistical inference under dependence, such as the block bootstrap (see, e.g., Carlstein 1986, Hall 1985, or Kunsch 1989), but they are not closely related to the methods that we develop.

There is a parallel between the technology developed here and that appropriate in the setting of nonparametric curve estimation from noisy data. This analogy has been discussed extensively by Diggle and Marron (1988), and so we shall not dwell on it here, except to note that literature in the curve estimation context distantly related to our own includes work of Hardle and Bowman (1988) and Hardle and Marron (1991) on bootstrap confidence bands for regression curves. The Poisson process context treated in this article is distinguished by, among other things, the need to develop a different approach to bootstrapping, with consequently different technology.

A major way in which our work differs from that of earlier authors is that we do not take a "Cox process" view of the intensity estimation problem. Earlier contributions (Diggle 1985; Diggle and Marron 1988) considered the intensity of the observed Poisson process to be a realization of a stationary stochastic process, and assessed performance of their estimation procedures by taking averages over that process as well as over the observed data conditional on intensity. This simplified their theoretical development considerably, because the estimation problem was transformed to one for stationary rather than "time-varying" processes. However, we feel that the Cox process view is much less appropriate in the confidence region problem than it was for point estimation, because it makes little sense to construct a confidence region for the average intensity. Therefore, we treat the Poisson process as genuinely nonstationary. This requires us to develop new theory for point estimation in this more complex case, complementing work of Diggle and Marron on point estimation, as well as contributing new results on confidence regions.

All of our work has a straightforward generalization to the case of multivariate Poisson processes, where intensity is a function from a multivariate Euclidean space to the set of positive numbers. The manner in which our univariate algorithms should be modified is so straightforward that it seems unnecessary to comment further. …