Local polynomial estimation of the intensity of a doubly stochastic Poisson process with bandwidth selection procedure
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We consider a doubly stochastic Poisson process with stochastic intensity λ_t =n q(X_t) where X is a continuous Itô semimartingale and n is an integer. Both processes are observed continuously over a fixed period [0,T]. An estimation procedure is proposed in a non parametrical setting for the function q on an interval I where X is sufficiently observed using a local polynomial estimator. A method to select the bandwidth in a non asymptotic framework is proposed, leading to an oracle inequality. If m is the degree of the chosen polynomial, the accuracy of our estimator over the Hölder class of order β is n^-β/2β+1 if m ≥β and it is optimal in the minimax sense if m ≥β. A parametrical test is also proposed to test if q belongs to some parametrical family. Those results are applied to French temperature and electricity spot prices data where we infer the intensity of electricity spot spikes as a function of the temperature.
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