Bernoulli

  • Bernoulli
  • Volume 18, Number 3 (2012), 1061-1098.

$\sqrt n$-consistent parameter estimation for systems of ordinary differential equations: bypassing numerical integration via smoothing

Shota Gugushvili and Chris A.J. Klaassen

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Abstract

We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We study the consistency of an alternative estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the right-hand side of the system applied to a nonparametrically estimated solution. This smooth and match estimator (SME) bypasses numerical integration altogether and reduces the amount of computational time drastically compared to ordinary least squares. Moreover, we show that under suitable regularity conditions this smooth and match estimation procedure leads to a $\sqrt n$-consistent estimator of the parameter of interest.

Article information

Source
Bernoulli, Volume 18, Number 3 (2012), 1061-1098.

Dates
First available in Project Euclid: 28 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1340887014

Digital Object Identifier
doi:10.3150/11-BEJ362

Mathematical Reviews number (MathSciNet)
MR2948913

Zentralblatt MATH identifier
1257.49033

Keywords
M-estimator $\sqrt n$-consistency nonparametric regression ODE system Priestley–Chao estimator

Citation

Gugushvili, Shota; Klaassen, Chris A.J. $\sqrt n$-consistent parameter estimation for systems of ordinary differential equations: bypassing numerical integration via smoothing. Bernoulli 18 (2012), no. 3, 1061--1098. doi:10.3150/11-BEJ362. https://projecteuclid.org/euclid.bj/1340887014


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