2008/01/21 by Patricia Reynaud-Bouret, Reynaud-Bouret, Patricia, Vincent Rivoirard +1
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.0801.3157
arxiv created 2008/01/21 · arxiv updated 2009/12/01
The purpose of this paper is to estimate the intensity of a Poisson process N by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of N with respect to ndx where n is a fixed parameter, is assumed to be non-compactly supported. The estimator fn,γ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of fn,γ. Then, minimax properties of fn,γ are established. We first prove that the rate of this estimator on Besov spaces \cal B^\alp,q when p≤ 2 is (ln(n)/n)\al/(1+2\al). This result has two consequences. First, it establishes that the minimax rate of Besov spaces \cal B^\alp,q with p≤ 2 when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. This result is new. Furthermore, fn,γ is adaptive minimax up to a logarithmic term. When p>2, the situation changes dramatically and the rate of fn,γ on Besov spaces \cal B^\alp,q is worse than (ln(n)/n)\al/(1+2\al). Finally, the random threshold depends on a parameter γ that has to be suitably chosen in practice. Some theoretical results provide upper and lower bounds of γ to obtain satisfying oracle inequalities. Simulations reinforce these results.