# Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme

vendredi 1 juin 2018, 9h30 - 10h30

Compound Poisson processes (CPPs) are the textbook example of pure jump Lévy processes (LPs). They have two defining parameters: the jump distribution $\mu$ and the intensity $\lambda$. A sample path is simply a staircase where the step size is $\mu$-distributed and the time between jumps follows an exponential distribution with parameter $\lambda$. Therefore, CPPs provide a simple, yet fundamental, model for random shocks in a system, which itself and its generalisations are applied in a myriad of applications within natural sciences, engineering and economics. In most of these, the underlying CPP is not perfectly observed: only $n$ discrete observations every $\Delta>0$ amount of time are available. Hence, the process may jump several times between two observations and we are effectively observing a random variable corrupted by a sum of a random number of copies of itself. Consequently, estimating the Lévy distribution $\lambda\mu$, or its density $\nu$ if it exists, is a non-linear statistical inverse problem.

In the last decade, this problem and its generalisation within LPs have attracted much attention (cf. [2]). Existing literature can be roughly split into high-frequency observations, $\Delta=\Delta_n\to 0$, and low-frequency, $\Delta$ fixed, and both regimes use different techniques to build estimators. We will present our recent results in [1], where we show that an estimator of $\nu$ constructed using the spectral approach (generally used in the second setting) is robust to both observation regimes and, under minimal tail assumptions, is minimax-optimal without knowledge of the Nikolskii-regularity of $\nu$ for the losses $L^p({R})$, $p\in[1,\infty]$. This is particularly novel because all existing results are shown either for $p=2$ or in settings where some $L^2$ structure can be exploited. Adaptive results are sparse and use model selection techniques that are especially well-suited for the $L^2$ setting. Instead, we use Lepskii’s method and, to do so, show new exponential-concentration inequalities. This includes one for the supremum of the fluctuations of the empirical characteristic function from which it follows that, up to logarithms, it concentrates at the parametric rate with sample size $T_{\lambda}:=\lambda \Delta n$; note that $T_{\lambda}$ is the expected number of jumps the CPP gives in $[0,\Delta n]$ and, thus, it is the natural sample size in this context.

References:

[1] Coca, A. J. (2018) Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme, ArXiv preprint, https://arxiv.org/abs/1803.09849

[2] Belomestny, D., Comte, F., Genon-Catalot, V., Masuda, H., Rei{\ss}, M. (2015) Lévy Matters IV, Springer.