# Infill asymptotics and bandwidth selection for kernel estimators of spatial intensity functions

## TL;DR

This paper analyzes the asymptotic mean squared error of kernel estimators for spatial intensity functions, deriving optimal bandwidths for both fixed and adaptive estimators under smoothness assumptions.

## Contribution

It establishes the asymptotic optimal bandwidth rates for kernel estimators and adaptive estimators of spatial intensity functions, extending existing theory.

## Key findings

- Optimal bandwidth for fixed kernel estimator: n^{-1/(d+4)}
- Optimal adaptive bandwidth: n^{-1/(d+8)}
- Theoretical foundation for adaptive kernel estimation in spatial processes

## Abstract

We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We show that when $n$ independent copies of a point process in $\mathbb R^d$ are superposed, the optimal bandwidth $h_n$ is of the order $n^{-1/(d+4)}$ under appropriate smoothness conditions on the kernel and true intensity function. We apply the Abramson principle to define adaptive kernel estimators and show that asymptotically the optimal adaptive bandwidth is of the order $n^{-1/(d+8)}$ under appropriate smoothness conditions.