Second order asymptotic efficiency for a Poisson process

TL;DR

This paper investigates the estimation of the mean function of a periodic inhomogeneous Poisson process, demonstrating classical efficiency and proposing an estimator with second order asymptotic efficiency over Sobolev ellipsoids.

Contribution

It introduces a new estimator that achieves second order asymptotic efficiency, extending classical results on efficiency for Poisson process mean estimation.

Findings

Empirical mean function attains the classical lower bound for MISE.

Proposed estimator is second order asymptotically efficient.

Analysis conducted over Sobolev ellipsoids following Pinsker's approach.

Abstract

We consider the problem of the estimation of the mean function of an inhomogeneous Poisson process when its intensity function is periodic. For the mean integrated squared error (MISE) there is a classical lower bound for all estimators and the empirical mean function attains that lower bound, thus it is asymptotically efficient. Following the ideas of the work by Golubev and Levit, we compare asymptotically efficient estimators and propose an estimator which is second order asymptotically efficient. Second order efficiency is done over Sobolev ellipsoids, following the idea of Pinsker.