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

2017/03/16 by Martin H. Kroll, Kroll, Martin
Engineering · Mathematics · Medicine · #60G55 #62G05 #FOS: Mathematics #Optical Imaging and Spectroscopy Techniques #Photoacoustic and Ultrasonic Imaging #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1703.05619

openalex publication_date 2017/03/16 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

We consider the nonparametric estimation of the intensity function of a\nPoisson point process in a circular model from indirect observations\nN₁,\…,N_n. These observations emerge from hidden point process\nrealizations with the target intensity through contamination with additive\nerror. In case that the error distribution can only be estimated from an\nadditional sample Y₁,\…,Y_m we derive minimax rates of convergence with\nrespect to the sample sizes n and m under abstract smoothness conditions\nand propose an orthonormal series estimator which attains the optimal rate of\nconvergence. The performance of the estimator depends on the correct\nspecification of a dimension parameter whose optimal choice relies on\nsmoothness characteristics of both the intensity and the error density. We\npropose a data-driven choice of the dimension parameter based on model\nselection and show that the adaptive estimator attains the minimax optimal\nrate.\n

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