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Local polynomial estimation of the intensity of a doubly stochastic Poisson process with bandwidth selection procedure
Thomas Deschatre · 2018-11-28 · via math.ST updates on arXiv.org

We consider a doubly stochastic Poisson process with stochastic intensity $λ_t =n q\left(X_t\right)$ where $X$ is a continuous Itô semimartingale and $n$ is an integer. Both processes are observed continuously over a fixed period $\left[0,T\right]$. 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 $n^{\frac{-β}{2β+1}}$ if $m \geq \lfloor β\rfloor$ and it is optimal in the minimax sense if $m \geq \lfloor β\rfloor$. 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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