Abstract
We consider a dynamical system with small noise for which the drift is parametrized by a finite dimensional parameter. For this model, we consider minimum distance estimation from continuous time observations under $l^{p}$-penalty imposed on the parameters in the spirit of the Lasso approach, with the aim of simultaneous estimation and model selection. We study the consistency and the asymptotic distribution of these Lasso-type estimators for different values of $p$. For $p=1,$ we also consider the adaptive version of the Lasso estimator and establish its oracle properties.
Highlights
Ordinary differential equation models are the result of averaging and/or neglecting some details of an original system without modeling a complex system with a huge number of degrees of freedom or tuning parameters
It is natural to think of the noise as small, for example when one is considering the dynamics of macroscopic quantities, i.e. averages of quantities of interest over a whole population or in the case of signal that travels through a perturbed medium, etcetera
What occurs for dynamical systems with small noise, is not so different from what happens in ordinary least squares (OLS) model estimation
Summary
Ordinary differential equation models are the result of averaging and/or neglecting some details of an original system without modeling a complex system with a huge number of degrees of freedom or tuning parameters. What occurs for dynamical systems with small noise, is not so different from what happens in ordinary least squares (OLS) model estimation. To introduce the idea of Lasso-type estimation we begin with linear models and OLS. In this framework model selection occurs when some of the regression parameters are estimated as zero. The estimators solutions to (1.1) are attractive because with them it is possible to perform estimation and model selection in a single step, i.e. the procedure does not need to estimate different models and compare them later with information criteria as the dimension of the space of the parameters does not change; just some of the components of the vector βj are assumed to be zero. We are able to prove that the adaptive estimation represents an oracle procedure
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