37 research outputs found
Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions
A truncated sequential procedure is constructed for estimating the drift
coefficient at a given state point based on discrete data of ergodic diffusion
process. A nonasymptotic upper bound is obtained for a pointwise absolute error
risk. The optimal convergence rate and a sharp constant in the bounds are found
for the asymptotic pointwise minimax risk. As a consequence, the efficiency is
obtained of the proposed sequential procedure.Comment: Published at http://dx.doi.org/10.3150/14-BEJ655 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models. *
In this paper we consider high dimensional ergodic diffusion models in nonparametric setting on the basis of discrete data, when the diffusion coefficients are unknown. For this problem, by using efficient sequential point-wise estimators we construct a model selection procedure and then we show sharp oracle inequalities, i.e. the inequalities in which the main term coefficient is closed to one. This means that the proposed sequential model selection procedure is optimal in this sense. Particularly, we show that the constructed procedure is the best in the class of weighted least square estimators with the Pinsker coefficients which provide the efficient estimation in the minimal asymptotical quadratic risk sense
Adaptive efficient analysis for big data ergodic diffusion models
We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions
General model selection estimation of a periodic regression with a Gaussian noise
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
Non asymptotic sharp oracle inequalities for high dimensional ergodic diffusion models. *
In this paper we consider high dimensional ergodic diffusion models in nonparametric setting on the basis of discrete data, when the diffusion coefficients are unknown. For this problem, by using efficient sequential point-wise estimators we construct a model selection procedure and then we show sharp oracle inequalities, i.e. the inequalities in which the main term coefficient is closed to one. This means that the proposed sequential model selection procedure is optimal in this sense. Particularly, we show that the constructed procedure is the best in the class of weighted least square estimators with the Pinsker coefficients which provide the efficient estimation in the minimal asymptotical quadratic risk sense
Uniform concentration inequality for ergodic diffusion processes observed at discrete times
AbstractIn this paper a concentration inequality is proved for the deviation in the ergodic theorem for diffusion processes in the case of discrete time observations. The proof is based on geometric ergodicity of diffusion processes. We consider as an application the nonparametric pointwise estimation problem of the drift coefficient when the process is observed at discrete times