ANZIAM J.
45 (2003), 91-114
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Numerical algorithms for constrained maximum likelihood estimation
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Z. F. Li
National Centre for Epidemiology
and Population Health
Australian National University
Canberra ACT 0200
Australia
zhengfeng.li@anu.edu.au
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M. R. Osborne
Centre for Mathematics and its Applications
School of Mathematical Sciences
Australian National University
Canberra ACT 0200
Australia
Mike.Osborne@maths.anu.edu.au
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Abstract
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This paper describes a SQP-type algorithm for
solving a constrained maximum likelihood
estimation problem that incorporates a number of
novel features. We call it MLESOL. MLESOL
maintains the use of an estimate of the Fisher
information matrix to the Hessian of the negative
log-likelihood but also encompasses a secant
approximation S
to the second-order part of the augmented
Lagrangian function along with tests for when to
use this information. The local quadratic model
used has a form something like that of Tapia's
SQP augmented scale BFGS secant method but
explores the additional structure of the
objective function. The step choice algorithm is
based on minimising a local quadratic model
subject to the linearised constraints and an
elliptical trust region centred at the current
approximate minimiser. This is accomplished using
the Byrd and Omojokun trust region approach,
together with a special module for assessing the
quality of the step thus computed. The numerical
performance of MLESOL is studied by means of an
example involving the estimation of a mixture
density.
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