Institut für Mathematik - Chairs - Mathematical Statistics

Mathematical statistics deal with the development of procedures, which allow for inference for a distribution based on random empirical data which are drawn from that distribution; e.g. in order to estimate certain properties of that distribution or to check some assumptions on the distribution. Furthermore, the quality of such procedures is investigated. For instance, this can be carried out by calculating the risk, i.e. the expected distance between the estimator and the true quantity to be estimated. In general, statisticians try to discover procedures which minimize that risk. Most frequently, this minimization is studied from an asymptotic viewpoint, i.e. the number of observations is assumed to tend to infinity. A major field of our research is nonparametric statistics where the target distribution is not uniquely described by finitely many real-valued parameters but must be modelled as a function (for example, density estimation and regression analysis). We focus on measurement error models where the data are contaminated by additional random effects. The reconstruction of the original target function represents a statistical inverse problem; in many cases it leads to a deconvolution problem.
Stochastic processes, i.e. the collection of several random variables, e.g. random functions, occur in many statistical problems. For instance, in some models, one can construct Ito processes, whose observation is asymptotically equivalent to some corresponding curve estimation problems (in LeCam's sense). Also, the data themselves can be modelled as stochastic processes when the true data are extremely high-dimensional. Moreover, many dependent financial data can be seen as a time series (e.g. GARCH models) so that statistical methods are applicable.