# Nonparametric intensity estimation from noisy observations of a Poisson process under unknown error distribution

## TL;DR

This paper develops a nonparametric method for estimating the intensity of a Poisson process from noisy, indirect observations, achieving minimax optimal rates even when the error distribution is unknown and estimated from additional data.

## Contribution

It introduces an orthonormal series estimator that adapts to unknown smoothness and error distribution, providing minimax optimal convergence rates in a circular Poisson process model.

## Key findings

- Estimator attains minimax optimal convergence rates.
- Data-driven dimension selection improves adaptivity.
- Method effectively handles unknown error distribution.

## Abstract

We consider the nonparametric estimation of the intensity function of a Poisson point process in a circular model from indirect observations $N_1,\ldots,N_n$. These observations emerge from hidden point process realizations with the target intensity through contamination with additive error. In case that the error distribution can only be estimated from an additional sample $Y_1,\ldots,Y_m$ we derive minimax rates of convergence with respect to the sample sizes $n$ and $m$ under abstract smoothness conditions and propose an orthonormal series estimator which attains the optimal rate of convergence. The performance of the estimator depends on the correct specification of a dimension parameter whose optimal choice relies on smoothness characteristics of both the intensity and the error density. We propose a data-driven choice of the dimension parameter based on model selection and show that the adaptive estimator attains the minimax optimal rate.