Equivalence of smoothing parameter selectors in density and intensity estimation

Kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary Poisson process. In each case the amount of smoothing, controlled by the bandwidth, that is, smoothing parameter, is crucial to the performance of the estimator. Bandwidth selection by cross-validation has been widely studied in the context of density estimation. A bandwidth selector in the intensity estimation case has been proposed that minimizes an estimate of the mean squared error under the assumption that the data are generated by a stationary Cox process. This article shows that these two methods each select the same bandwidth, even though they are motivated in much different ways. In addition to providing further justification of each method, this equivalence of smoothing parameter selectors yields new insights for both density and intensity estimation. A benefit for intensity estimation is that this equivalence makes it clear how the Cox process method may be applied to kernels that are nonuniform, or even of higher order. Another benefit is that this duality between problems makes it clear how to apply the well-developed asymptotic methods for understanding density estimation in the intensity setting. A benefit for density estimation is that it motivates an analog of the Cox process method, which provides a useful nonasymptotic means of studying that problem. The specific forms of the estimators and smoothing parameter selectors are introduced in Section 1. The basic equivalence result is stated in Section 2. Sections 3 and 4 describe new insights that follow for intensity and density estimation, respectively. Section 5 discusses modification of these ideas to take boundary effects into consideration and shows how they can be used to motivate new boundary adjustments in intensity estimation.