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Numerical Methods for Nonlinear Estimating Equations$
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Christopher G. Small and Jinfang Wang

Print publication date: 2003

Print ISBN-13: 9780198506881

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506881.001.0001

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Root selection and dynamical systems

Root selection and dynamical systems

(p.241) 7 Root selection and dynamical systems
Numerical Methods for Nonlinear Estimating Equations

Christopher G. Small

Jinfang Wang

Oxford University Press

This chapter explores the relationship between the numerical methods described in earlier chapters and the theory of dynamical systems. An estimating function defines a dynamical estimating system, whose domains of attraction and repulsion can be studied in relation to the estimation problem. For instance, stability of roots to an estimating equation can be studied using the linearization method or the Liapunov's method. In particular, it can be shown that a consistent root of an estimating equation is an asymptotically stable fixed point of the associated dynamical estimating system. The Newton-Raphson method is reexamined in detail from the perspective of the theory of dynamical systems, and derivations of and formal proofs for the properties of the modified Newton's methods are given. This chapter also explores the Julia sets and domains of attraction of estimating functions, taking the estimation of the correlation coefficient for bivariate normal data as an example.

Keywords:   asymptotically stable fixed point, correlation coefficient, domain of attraction, domain of repulsion, dynamical estimating system, dynamical system, Julia set, Liapunov's method, linearization method, stability of root

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