The concept of distance is ancient and ubiquitous. Being able to quantify the distance between objects has important applications everywhere in mathematics; just think of approximation theory, information theory, optimal transport, etc…
In this talk we will focus on the Le Cam distance, a (pseudo-)metric on the class of statistical experiments, and we will discuss what kind of information can be extracted from a control of such a distance.
As a driving example, we consider the problem of estimating the Lévy density $f$ of a pure jump Lévy process, possibly of infinite variation, from the high frequency observation of one trajectory. We discuss two different approaches and highlight the connections among them.
The first one consists in reducing the problem of the nonparametric estimation of $f$ to an easier one, namely the estimation of a drift of a Gaussian white noise model.
More precisely, we establish a global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a Lévy process and a Gaussian white noise experiment observed up to a time $T$, with $T$ tending to $\infty$. These approximations are given in the sense of the Le Cam distance, under some smoothness conditions on the unknown Lévy density. The asymptotic equivalences are established by constructing explicit equivalence mappings that can be used to reproduce one experiment from the other and to transfer estimators.
The second approach consists in directly constructing an estimator of the Lévy density. For that we use a compound Poisson approximation and we build a linear wavelet estimator. Its performance is studied in terms of $L_p$ loss functions, $p\geq1$, over Besov balls. The resulting rates are minimax-optimal for a large class of Lévy processes. This work was done in collaboration with Céline Duval (MAP5, Paris).