Discretized nonparametric regression for functional data

FORZANI, LILIANA; FRAIMAN, RICARDO; LLOP, PAMELA

Cartagena de Indias

Congreso; XIII Congreso Latinoam de Probabilidad y Estadística Matemática; 2014

SLAPEM

Technological progress in collecting and storing data has provide data sets recorded at finite grids of points which, thanks to the new technologies, become denser and denser over time. Although in practice data always come in form of finite dimensional vectors, from the theoretical point of view, the classic multivariate techniques are not suitable to deal with this kind of data. In this direction, the asymptotic theory can be analyzed either assuming the existence of continuous underlying stochastic processes ideally observed at every point, or transforming the (observed) discrete values into a functions via interpolation (errorless case), smoothing (if error is present), splines or series approximations. When dealing with the regression problem for discretized functional data, a natural question that emerges is which the relationship between the ``ideal'' nonparametric regression estimate computed with the entire curve and the one computed with the discretized sample. In this direction, we state conditions under which the consistency of the estimator computed with the discretized trajectories can be derived from the consistency of the one based on the whole curves. Also, we give conditions on the grid size discretization in order to achieve the same rates of convergence that in infinite dimensional setting. Those results are consequence of two more general results which, besides discretization, also includes the case of smoothing via regularization, basis representation or interpolation data.