Adaptive thresholding estimation of a Poisson intensity with infinite support

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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 Bₚ,q when p≤ 2 is ((n)/n)/(1+2). This result has two consequences. First, it establishes that the minimax rate of Besov spaces Bₚ,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 Bₚ,q is worse than ((n)/n)/(1+2). 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.