Threshold Estimation for Stochastic Processes with Small Noise

    Research output: Contribution to journalArticle

    Abstract

    Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.

    Original languageEnglish
    Pages (from-to)951-988
    Number of pages38
    JournalScandinavian Journal of Statistics
    Volume44
    Issue number4
    DOIs
    Publication statusPublished - 2017 Dec 1

    Fingerprint

    Least Squares Estimator
    Stochastic Processes
    Filter
    Discrete Observations
    Jump Process
    Asymptotically equivalent
    Justify
    Standard deviation
    Asymptotic distribution
    Stochastic Equations
    Numerical Study
    Shock
    Unstable
    Differential equation
    Estimator
    Unknown
    Threshold estimation
    Stochastic processes
    Least squares estimator

    Keywords

    • 60G52
    • 60J75
    • drift estimation
    • mighty convergence
    • semimartingale noise
    • small noise asymptotics
    • stochastic differential equation
    • threshold estimator MSC2010:62F12; 62M05

    ASJC Scopus subject areas

    • Statistics and Probability
    • Statistics, Probability and Uncertainty

    Cite this

    Threshold Estimation for Stochastic Processes with Small Noise. / Shimizu, Yasutaka.

    In: Scandinavian Journal of Statistics, Vol. 44, No. 4, 01.12.2017, p. 951-988.

    Research output: Contribution to journalArticle

    @article{7977babe373b46a1a136dd1e242f217f,
    title = "Threshold Estimation for Stochastic Processes with Small Noise",
    abstract = "Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a L{\'e}vy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.",
    keywords = "60G52, 60J75, drift estimation, mighty convergence, semimartingale noise, small noise asymptotics, stochastic differential equation, threshold estimator MSC2010:62F12; 62M05",
    author = "Yasutaka Shimizu",
    year = "2017",
    month = "12",
    day = "1",
    doi = "10.1111/sjos.12287",
    language = "English",
    volume = "44",
    pages = "951--988",
    journal = "Scandinavian Journal of Statistics",
    issn = "0303-6898",
    publisher = "Wiley-Blackwell",
    number = "4",

    }

    TY - JOUR

    T1 - Threshold Estimation for Stochastic Processes with Small Noise

    AU - Shimizu, Yasutaka

    PY - 2017/12/1

    Y1 - 2017/12/1

    N2 - Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.

    AB - Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.

    KW - 60G52

    KW - 60J75

    KW - drift estimation

    KW - mighty convergence

    KW - semimartingale noise

    KW - small noise asymptotics

    KW - stochastic differential equation

    KW - threshold estimator MSC2010:62F12; 62M05

    UR - http://www.scopus.com/inward/record.url?scp=85025076407&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85025076407&partnerID=8YFLogxK

    U2 - 10.1111/sjos.12287

    DO - 10.1111/sjos.12287

    M3 - Article

    AN - SCOPUS:85025076407

    VL - 44

    SP - 951

    EP - 988

    JO - Scandinavian Journal of Statistics

    JF - Scandinavian Journal of Statistics

    SN - 0303-6898

    IS - 4

    ER -