Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T10:18:32.329Z Has data issue: false hasContentIssue false
  • English
  • Français

Scoring with constraints

Published online by Cambridge University Press:  17 February 2009

Michael R. Osborne
Michael R. Osborne
Affiliation:
Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper considers the solution of estimation problems based on the maximum likelihood principle when a fixed number of equality constraints are imposed on the parameters of the problem. Consistency and the asymptotic distribution of the parameter estimates are discussed as n → ∞, where n is the number of independent observations, and it is shown that a suitably scaled limiting multiplier vector is known. It is also shown that when this information is available then the good properties of Fisher's method of scoring for the unconstrained case extend to a class of augmented Lagrangian methods for the constrained case. This point is illustrated by means of an example involving the estimation of a mixture density.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 1 , July 2000 , pp. 9 - 25
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Ascher, U., Mattheij, R., and Russell, R. D., The Numerical Solution of Boundary Value Problems (Birkhauser, 1988.)Google Scholar
[2]Ascher, U. and Osborne, M. R., “A note on solving nonlinear equations and the natural criterion function”, J.O.T.A 55 (1988) 147–152.CrossRefGoogle Scholar
[3]Childs, S. B. and Osborne, M. R., “Fitting solutions of ordinary differential equations to observed data”, TAMU Technical Report CS94–071 (1994).Google Scholar
[4]Fisher, N. I., Lewis, T., and Embleton, B. J. J., Statistical Analysis of Spherical Data (Chapman and Hall, 1992).Google Scholar
[5]Fletcher, R., Practical Methods of Optimization, Vol.2: Constrained Optimization (Wiley, 1981).Google Scholar
[6]Osborne, M. R., “Fisher's method of scoring”, Int. Stat. Rev. 60 (1992) 99–117.CrossRefGoogle Scholar
[7]Osborne, M. R., “Cyclic reduction, dichotomy, and the estimation of differential equations”, J. Comp. and Appl. Math. 86 (1997) 271–286.CrossRefGoogle Scholar
[8]Powell, M. J. D. and Reid, J. K., “On applying Householder's method to linear least squares problems”, Proc. IFIP Congress (1968) 122–126.Google Scholar
[9]Sen, P. K. and Singer, J. M., Large Sample Methods in Statistics (Chapman and Hall, 1993).CrossRefGoogle Scholar
[10]Shuster, J. J. and Downing, D. J., “Two-way contingency tables for complex sampling schemes”, Biometrica 63 (1976) 271–276.CrossRefGoogle Scholar