Least Squares Regression for Non stationary Designs 27 10 2017

Least Squares Regression for Non-Stationary Designs. David Barrera. Postdoc, École Polytechnique, Palaiseau, France. Escuela de Ciencias Universidad EAFIT. Octubre 27 de 2017.
Seminario de Doctorado en Ingeniería Matemática Universidad EAFIT
Seminar of the PhD in Mathematical Engineering EAFIT University.
Abstract: The main goal of this talk is to present a series of new results concerning the rate of convergence for least square estimates in the case in which the sampling process (the « design ») is not i.i.d. Our results are given in the setting of nonparametric regression, and they cover the corresponding estimates in the i.i.d. case (as given, for instance, in [1]) without any essential loss in the respective rates of convergence nor the introduction of additional hypotheses in order to carry out the proofs. They justify also a more general -but very natural- interpretation of the least-squares regression function as the “best” approximation to the response function in a given statistical experiment, and provide further theoretical ground for the research on numerical methods in which a non-stationary evolution has to be considered.
We illustrate these results and their aforementioned interpretation in the numerical context by looking at estimation problems in which the i.i.d. setting is either not satisfiable or not convenient, emphasizing in particular the Markovian setting. We also illustrate
the relevance of these tools in the error analysis of Monte Carlo algorithms like the one in [2].
[1] Györfi, L.; Kohler,M.; Krzyzak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression. Springer Ser. Statist.
[2] Fort, G.; Gobet, E. and Moulines, E. (2017) MCMC Design-Based Non-Parametric Regression for Rare Event. Application for Nested Risk Computations. To appear in Monte Carlo Methods Appl.
* This talk was first given during the 11th International Conference on Monte Carlo Methods and Applications, held at HEC Montréal
(Canada) on the days July 3-7, 2017.

Regresión de mínimos cuadrados para diseños no estacionarios