Efficient estimation for diffusions sampled at high frequency over a fixed time interval

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Efficient estimation for diffusions sampled at high frequency over a fixed time interval. / Jakobsen, Nina Munkholt; Sørensen, Michael.

In: Bernoulli, Vol. 23, No. 3, 01.08.2017, p. 1874-1910.

Research output: Contribution to journalJournal article – Annual report year: 2017Researchpeer-review

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@article{0a595679463a4235bb2b9bb1634771ac,
title = "Efficient estimation for diffusions sampled at high frequency over a fixed time interval",
abstract = "Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model.",
keywords = "Approximate martingale estimating functions, Discrete time sampling of diffusions, In-fill asymptotics, Normal variance-mixtures, Optimal rate, Random Fisher information, Stable convergence, Stochastic differential equation",
author = "Jakobsen, {Nina Munkholt} and Michael S{\o}rensen",
year = "2017",
month = "8",
day = "1",
doi = "10.3150/15-BEJ799",
language = "English",
volume = "23",
pages = "1874--1910",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "Bernoulli Society for Mathematical Statistics and Probability",
number = "3",

}

RIS

TY - JOUR

T1 - Efficient estimation for diffusions sampled at high frequency over a fixed time interval

AU - Jakobsen, Nina Munkholt

AU - Sørensen, Michael

PY - 2017/8/1

Y1 - 2017/8/1

N2 - Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model.

AB - Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model.

KW - Approximate martingale estimating functions

KW - Discrete time sampling of diffusions

KW - In-fill asymptotics

KW - Normal variance-mixtures

KW - Optimal rate

KW - Random Fisher information

KW - Stable convergence

KW - Stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=85016166903&partnerID=8YFLogxK

U2 - 10.3150/15-BEJ799

DO - 10.3150/15-BEJ799

M3 - Journal article

VL - 23

SP - 1874

EP - 1910

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 3

ER -