General model selection estimation of a periodic regression with a Gaussian noise

TL;DR

This paper introduces a model selection method for estimating periodic functions in continuous-time regression with Gaussian noise, providing risk bounds and extending to discrete data scenarios.

Contribution

It proposes a noise-agnostic model selection procedure with non-asymptotic risk bounds and demonstrates its advantages over least squares estimates.

Findings

Risk bounds are derived for the estimator.

The method is asymptotically minimax.

Extension to discrete data is achieved.

Abstract

This paper considers the problem of estimating a periodic function in a continuous time regression model with an additive stationary gaussian noise having unknown correlation function. A general model selection procedure on the basis of arbitrary projective estimates, which does not need the knowledge of the noise correlation function, is proposed. A non-asymptotic upper bound for quadratic risk (oracle inequality) has been derived under mild conditions on the noise. For the Ornstein-Uhlenbeck noise the risk upper bound is shown to be uniform in the nuisance parameter. In the case of gaussian white noise the constructed procedure has some advantages as compared with the procedure based on the least squares estimates (LSE). The asymptotic minimaxity of the estimates has been proved. The proposed model selection scheme is extended also to the estimation problem based on the discrete data applicably to the situation when high frequency sampling can not be provided.