Simultaneous White Noise Models and Optimal Recovery of Functional Data

We consider i.i.d. realizations of a Gaussian process on $[0,1]$ satisfying prescribed regularity conditions. The data consist of discrete samplings of these realizations in i.i.d. Gaussian noise and the goal is estimation of the underlying trajectories. Further, we want our estimates to enjoy expected $L_2$ errors, conditioned on the realized trajectories, which attain optimal rates. Under general conditions on both design and process, an asymptotic equivalence, in Le Cam's sense, is established between an experiment which simultaneously describes these realizations and a collection of white noise models. The risk properties of our estimation goal may then be studied in an idealized setting and benchmarks established for practically implementable procedures. In this context, the white noise models are projected onto a basis satisfying general conditions in relation to the covariance kernel of the process which generated the data. This reduces the problem of initial interest to that of recovering a collection of normal means in euclidean norm, the means of interest having Gaussian structure. A variant of Stein estimation is applied for recovery of these means and a key inequality derived showing that the corresponding risks, conditioned on the underlying means, can be made arbitrarily close to those that an oracle with knowledge of the process would attain. This establishes various notions of optimality for our recovery procedure. Finally, guarantees are derived for practically implementable variants and empirical performance is illustrated through simulated and real data examples.Ph.D