2008/10/29 by Patricia Reynaud-Bouret, Reynaud-Bouret, Patricia, Vincent Rivoirard +1 · 1 citation
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.0810.5204
Submitted to the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)
arxiv created 2008/10/29 · 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 possible logarithmic term. Then, minimax properties of fn,γ on Besov spaces \cal B\ensuremath αp,q are established. Under mild assumptions, we prove that sup_f∈ B\ensuremath αp,q∩ \ensuremath \mathbb L∞ \ensuremath \mathbb E(\ensuremath | | fn,γ-f| |22)≤ C((log n)/(n))^\frac\ensuremath α\ensuremath α+1/2+(1/2-(1)/(p))+ and the lower bound of the minimax risk for \cal B\ensuremath αp,q∩ \ensuremath \mathbb L∞ coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces \cal B\ensuremath αp,q with p≤ 2 when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When p>2, the rate exponent, which depends on p, deteriorates when p increases, which means that the support plays a harmful role in this case. Furthermore, fn,γ is adaptive minimax up to a logarithmic term.