Asymptotic theory of semiparametric z-estimators for stochastic processes with applications to ergodic diffusions and time series

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3 Citations (Scopus)

Abstract

This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψn(θ, ĥn) = 0 with an abstract nuisance parameter h when the compensator of ψn is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

Original languageEnglish
Pages (from-to)3555-3579
Number of pages25
JournalAnnals of Statistics
Volume37
Issue number6 A
DOIs
Publication statusPublished - 2009 Dec
Externally publishedYes

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Nuisance Parameter
Asymptotic Theory
Stochastic Processes
Time series
Estimator
Ergodic Processes
Asymptotic Efficiency
Semimartingale
Estimating Equation
Empirical Process
Infinite-dimensional Spaces
Time Series Models
Diffusion Model
Compensator
Coefficient
Asymptotic Normality
Smooth function
Diffusion Process
Diffusion Coefficient
Process Model

Keywords

  • Asymptotic efficiency
  • Discrete observation
  • Ergodic diffusion
  • Estimating function
  • Metric entropy
  • Nuisance parameter

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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title = "Asymptotic theory of semiparametric z-estimators for stochastic processes with applications to ergodic diffusions and time series",
abstract = "This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψn(θ, ĥn) = 0 with an abstract nuisance parameter h when the compensator of ψn is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.",
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N2 - This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψn(θ, ĥn) = 0 with an abstract nuisance parameter h when the compensator of ψn is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

AB - This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψn(θ, ĥn) = 0 with an abstract nuisance parameter h when the compensator of ψn is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

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