Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities

Abstract

In this paper, we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. We study consequently the adaptive properties of penalized projection estimators. At first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants. The crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of Poisson processes which are analogous to Talagrand's inequalities for empirical processes.

Nonparametric Estimation for Lévy Models Based on Discrete-Sampling

“…For instance, Ibragimov and Has'minskii [19] and Barron et al [2] provided this kind of asymptotics for the problem of density estimation based on i.i.d. random variables, while Kutoyants [22] and Reynaud-Bouret [25] considered the problem of intensity estimation of a finite Poisson point processes. This last set-up is relevant for our problem since the jumps of a Lévy process can be associated with a (possibly infinite) Poisson point process on R + × R\{0} (see e.g.…”

Concentration inequalities, counting processes and adaptive statistics

“…This result is an adapted and simpler version of the one presented in [41], which can be extended to other settings than just histograms.…”

“…families with complex cardinality: there are more models with the same dimension D than a power of D). Indeed, there exists a minimax lower bound (see Proposition 4 of [41]) that proves the existence of this logarithmic factor. See also [5] and [6] for a more thorough study in the Gaussian setup.…”

“…This section is mainly inspired by [41], but we give a simpler version here. We observe a Poisson process N , on X (say a compact subset of R d ) and we want to estimate its intensity s with respect to the Lebesgue measure, denoted µ.…”

The relationship between intensity and stochastic matrices for continuous-time discrete value stochastic non-homogeneous processes with Markov property

“…Non-homogeneous (sometimes called inhomogeneous) Markov stochastic processes often appear in scientific literature as solutions of many real problems due to twenty-four hour or seasonal fluctuations of probability of many real events [2], [4], [5], [6], [10], [11], [12], [14], [17]. But the general theory of these processes very seldom appears in textbooks of stochastic processes.…”