Statistical inference across scales.

Summary This thesis deals with the cross-scale estimation problem for a stochastic process. We study how the choice of the sampling step impacts the statistical procedures. We are interested in the estimation of jump processes from the observation of a discretized trajectory on [0, T]. When the length of the observation interval T goes to infinity, the sampling step tends either to 0 (microscopic scale), to a positive constant (intermediate scale) or to infinity (macroscopic scale). In each of these regimes we assume that the number of observations tends to infinity. First, the particular case of a compound Poisson process of unknown intensity with symmetric jumps {-1,1} is studied. Chapter 2 illustrates the notion of statistical estimation in the three scales defined above. In this model, we are interested in the properties of statistical experiments. We show the property of Local Asymptotic Normality in the three microscopic, intermediate and macroscopic scales. The Fisher information is then known for each of these regimes. Then we analyze how an intensity estimation procedure that is efficient (minimum variance) at a given scale behaves when applied to observations from a different scale. We look at the estimator of the empirical quadratic variation, which is efficient in the macroscopic regime, and we use it on data coming from the intermediate or microscopic regimes. This estimator remains efficient in the microscopic scales, but shows a substantial loss of information at intermediate scales. A unified estimation procedure is proposed, which is efficient in all regimes. Chapters 3 and 4 study the nonparametric estimation of the jump density of a compound renewal process in the microscopic regimes, when the sampling step tends to 0. An estimator of this density using wavelet methods is constructed. It is adaptive and minimax for sampling steps that decrease in T^{-alpha}, for alpha>0. The estimation procedure relies on the inversion of the composition operator giving the law of increments as a nonlinear transformation of the law of jumps that one seeks to estimate. The inverse operator is explicit in the case of the compound Poisson process (Chapter 3), but has no analytical expression for compound renewal processes (Chapter 4). In the latter case, it is approximated via a fixed point technique. Chapter 5 studies the problem of loss of identifiability in macroscopic regimes. If a jump process is observed with a large sampling step, some boundary approximations, such as the Gaussian approximation, become valid. This can lead to a loss of identifiability of the law that generated the process, when its structure is more complex than the one studied in Chapter 2. In a first step, a toy model with two parameters is considered. Two different regimes emerge from the study: a regime where the parameter is no longer identifiable and one where it remains identifiable but where the optimal estimators converge with slower speeds than the usual parametric speeds. From the particular case study, we derive lower bounds showing that there is no convergent estimator for pure jump Lévy processes or for compound renewal processes in macroscopic regimes where the sampling step grows faster than the root of T. Finally we identify macroscopic regimes where the increments of a compound Poisson process are indistinguishable from Gaussian random variables, and regimes where there is no convergent estimator for compound Poisson processes depending on too many parameters.