Concentration inequalities, counting processes and adaptive statistics

Abstract

Abstract.Adaptive statistics for counting processes need particular concentration inequalities to define and calibrate the methods as well as to precise the theoretical performance of the statistical inference. The present article is a small (non exhaustive) review of existing concentration inequalities that are useful in this context. Résumé. Les statistiques adaptatives pour les processus de comptage nécessitent des inégalités deconcentration particulières pour définir et calibrer les méthodes ainsi que pour comprendre les performances de l'inférence statistique. Cet article est une revue non exhaustive des inégalités de concentration qui sont utiles dans ce contexte.Mathematics Subject Classification: 62G05, 62G10, 62M07, 62M09, 60G55, 60E15.

Adaptive estimation of intensity in a doubly stochastic Poisson process

“…The derivation of the different proofs requires the two concentration inequalities from Reynaud‐Bouret (2014) and Houdré and Reynaud‐Bouret (2003). The concentration inequality of Proposition 6 is a weak Bernstein inequality; the other of Proposition 7 is an inequality for the Poisson U‐statistic.…”

“…Proposition (Reynaud‐Bouret, 2014, equation (2.2)) Let $$$ f:\mathbb{R}\to \mathbb{R} $$$ be a Borel function. Let $$$ N $$$ be an inhomogeneous Poisson process with intensity $$$ {\lambda}_{\cdotp } $$$, $$$ {\Lambda}_{\cdotp }={\int}_0^{\cdotp }{\lambda}_u du $$$ and $$$ M=N-\Lambda $$$.…”

“…In this equivalent framework, the behavior of $$$ \tilde{X} $$$ then is not affected by the growing time horizon while the number of jumps increases. This equivalent framework is considered in (Reynaud‐Bouret, 2003, 2014) who estimate the intensity of an inhomogeneous Poisson process. The statistical setting is also the same as observing $$$ n $$$ i.i.d.…”