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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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Working with roots

Working with roots

(p.74) 4 Working with roots
Numerical Methods for Nonlinear Estimating Equations

Christopher G. Small

Jinfang Wang

Oxford University Press

This chapter studies the broad class of models which have difficult problems of nonlinearity and multiple roots. These models include the mixture models, a model for bivariate normal paired data with standardized marginals, the stratified normal with common mean, regression with measurement error, weighted likelihood method, the Cauchy distribution, the symmetric stable laws, and examples with inconsistent global maximum likelihood estimates. The Tobit model is used to illustrate how one may determine if an estimating equation will have multiple solutions. The difficult issue of finding all possible roots is also discussed. This chapter also shows that multiple roots may be treated as a point process whose intensity may be assessed in the parameter space. Finally, smoothing is introduced as a technique to modify the estimating function or the artificial likelihood so as to remove unstable extraneous roots.

Keywords:   Cauchy distribution, inconsistent maximum likelihood estimate, mixture models, point assess, regression with measurement error, smoothing, stratified normal model, symmetric stable law, Tobit model, weighted likelihood

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