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 ℒ2-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.
| Original language | English |
|---|---|
| Pages (from-to) | 1083-1111 |
| Number of pages | 29 |
| Journal | Annals of the Institute of Statistical Mathematics |
| Volume | 62 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - Dec 2010 |
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Keywords
- Improved estimation
- Model selection procedure
- Non-parametric regression
- Oracle inequality
- Periodic regression
ASJC Scopus subject areas
- Statistics and Probability
Cite this
General model selection estimation of a periodic regression with a Gaussian noise. / Konev, Victor; Pergamenchtchikov, Serguei.
In: Annals of the Institute of Statistical Mathematics, Vol. 62, No. 6, 12.2010, p. 1083-1111.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - General model selection estimation of a periodic regression with a Gaussian noise
AU - Konev, Victor
AU - Pergamenchtchikov, Serguei
PY - 2010/12
Y1 - 2010/12
N2 - 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 ℒ2-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.
AB - 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 ℒ2-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.
KW - Improved estimation
KW - Model selection procedure
KW - Non-parametric regression
KW - Oracle inequality
KW - Periodic regression
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UR - http://www.scopus.com/inward/citedby.url?scp=77957882886&partnerID=8YFLogxK
U2 - 10.1007/s10463-008-0193-1
DO - 10.1007/s10463-008-0193-1
M3 - Article
VL - 62
SP - 1083
EP - 1111
JO - Annals of the Institute of Statistical Mathematics
JF - Annals of the Institute of Statistical Mathematics
SN - 0020-3157
IS - 6
ER -