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

Victor Konev, Serguei Pergamenchtchikov

    Research output: Contribution to journalArticle

    6 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)1083-1111
    Number of pages29
    JournalAnnals of the Institute of Statistical Mathematics
    Volume62
    Issue number6
    DOIs
    Publication statusPublished - Dec 2010

    Fingerprint

    Gaussian Noise
    Model Selection
    Correlation Function
    Regression
    Minimaxity
    Oracle Inequalities
    Upper bound
    Least Squares Estimate
    Discrete Data
    Continuous-time Model
    Nuisance Parameter
    Selection Procedures
    Gaussian White Noise
    Periodic Functions
    Estimate
    Regression Model
    Unknown
    Arbitrary
    Knowledge

    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 journalArticle

    Konev, Victor ; Pergamenchtchikov, Serguei. / General model selection estimation of a periodic regression with a Gaussian noise. In: Annals of the Institute of Statistical Mathematics. 2010 ; Vol. 62, No. 6. pp. 1083-1111.
    @article{0c231632a8a949c18b28f26ed4cb56ca,
    title = "General model selection estimation of a periodic regression with a Gaussian noise",
    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.",
    keywords = "Improved estimation, Model selection procedure, Non-parametric regression, Oracle inequality, Periodic regression",
    author = "Victor Konev and Serguei Pergamenchtchikov",
    year = "2010",
    month = "12",
    doi = "10.1007/s10463-008-0193-1",
    language = "English",
    volume = "62",
    pages = "1083--1111",
    journal = "Annals of the Institute of Statistical Mathematics",
    issn = "0020-3157",
    publisher = "Springer Netherlands",
    number = "6",

    }

    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

    UR - http://www.scopus.com/inward/record.url?scp=77957882886&partnerID=8YFLogxK

    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 -