Published

2007-01-01

UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS

AN INTRODUCTION TO OPTIMAL DESIGNS

Keywords:

función de información, matriz de información, criterios de optimalidad, teoremas de equivalencia, modelos de regresión no lineal (es)
Information function, Information matrix, Optimality criteria, Equivalence theorems, Nonlinear regression models (en)

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Authors

  • Víctor Ignacio López Universidad Nacional de Colombia Sede Medellín / Centro de Investigación en Matemáticas (CIMAT)
  • Rogelio Ramos Centro de Investigación en Matemáticas (CIMAT)
Introducimos varios conceptos utilizados en la teoría de diseños de experimentos óptimos. Definimos criterios de optimalidad utilizados en esta área y exploramos sus propiedades. Se listan algunos resultados importantes para encontrar diseños óptimos para modelos lineales y no lineales, entre ellos teoremas de equivalencia. Finalmente se presentan algunos ejemplos típicos donde se aplica la teoría vista anteriormente.
We introduce several concepts used in optimal experimental design. Optimality criteria used in this area are defined and their properties are explored. Some important results for finding optimal designs in linear and nonlinear models are listed, specially equivalence theorems are formulated. Finally, we present some examples where that theory is applied.

Una introducción a los diseños óptimos

An Introduction to Optimal Designs

VÍCTOR IGNACIO LÓPEZ1, ROGELIO RAMOS2

1Escuela de Estadística, Universidad Nacional de Colombia, Medellín. Profesor asistente. Estudiante de doctorado en Ciencias con Orientación en Probabilidad y Estadística. E-mail: vilopez@unalmed.edu.co
2Centro de Investigación en Matemáticas (CIMAT), Guanajuato, Gto., México. Investigador titular. E-mail: rramosq@cimat.mx

Resumen

Introducimos varios conceptos utilizados en la teoría de diseños de experimentos óptimos. Definimos criterios de optimalidad utilizados en esta área y exploramos sus propiedades. Se listan algunos resultados importantes para encontrar diseños óptimos para modelos lineales y no lineales, entre ellos teoremas de equivalencia. Finalmente se presentan algunos ejemplos típicos donde se aplica la teoría vista anteriormente.

Palabras clave: función de información, matriz de información, criterios de optimalidad, teoremas de equivalencia, modelos de regresión no lineal.

Abstract

We introduce several concepts used in optimal experimental design. Optimality criteria used in this area are defined and their properties are explored. Some important results for finding optimal designs in linear and nonlinear models are listed, specially equivalence theorems are formulated. Finally, we present some examples where that theory is applied.

Key words: Information function, Information matrix, Optimality criteria, Equivalence theorems, Nonlinear regression models.

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Referencias

1. Atkinson, A. C. (1996), ‘The Uselfulness of Optimum Experimental Designs’, Journal of the Royal Statistical Society. Series B (Methodological) 58(1), 59–76.

2. Atkinson, A. C. & Cox, D. R. (1974), ‘Planning Experiments for Discriminating Between Models’, Journal of the Royal Statistical Society. Series B (Methodological) 36(3), 321–348.

3. Atkinson, A. C. & Donev, A. N. (1992), Optimum Experimental Designs, Oxford Science Publications, New York.

4. Atkinson, A. C. & Fedorov, V. V. (1975), ‘Optimal Design: Experiments for Discriminating Between Several Models’, Biometrika 62(2), 289–303.

5. Biedermann, S., Dette, H. & Pepelyshev, A. (2005), ‘Optimal Discrimination Designs for Exponential Regression Models’, Preprint.

6. Biswas, A. & Chaudhuri, P. (2002), ‘An Efficient Design for Model Discrimination on Parameter Estimation in Linear Models’, Biometrika 89(3), 709–718.

7. Brown, L. D., Olkin, I., Sacks, J. & Wynn, H. P. (1985), Jack Karl Kiefer Collected Papers III, Design of Experiments, Springer Verlag, New York.

8. Chernoff, H. (1953), ‘Locally Optimal Designs for Estimating Parameters’, The Annals of Mathematical Statistics 24(24), 586–602.

9. Dette, H., Haines, L. M. & Imhof, L. A. (2003), ‘Maximin and Bayesian Optimal Designs for Regression Models’, Preprint. pp. 1-15.

10. Dette, H., Melas, V. B. & Pepelyshev, A. (2004), ‘Optimal Designs for a Class of Nonlinear Regression Models’, The Annals of Statistics 32(5), 2142–2167.

11. Dette, H., Melas, V. B. & Wong, W. K. (2005), ‘Optimal Design for Goodness-of-Fit of the Michaelis-Menten Enzyme Kinetic Function’, Journal of the American Statistical Association 100(472), 1370–1381.

12. Ford, I., Tornsney, B. & Wu, C. F. J. (1992), ‘The Use of a Canonical Form in the Construction of Locally Optimal Designs for Nonlinear Problems’, Journal of the Royal Statistical Society, Series B (Methodological) 54.

13. Kiefer, J. (1959), ‘Optimum Experimental Designs (With Discussion)’, Journal Royal Statistical Society, B 21, 272–319.

14. Kiefer, J. & Wolfowitz, J. (1960), ‘The Equivalence of Two Extremum Problems’, Canadian Journal of Mathematics 12, 363–366.

15. Pukelsheim, F. (1993), Optimal Design of Experiments, John Wiley & Sons, New York.

16. Pukelsheim, F. & Rosenberger, J. L. (1993), ‘Experimental Designs for Model Discrimination’, Journal of the American Statistical Association 88(422), 642–649.

17. Pukelsheim, F. & Torsney, B. (1991), ‘Optimal Weights for Experimental Designs on Linearly Independent Support Points’, The Annals of Statistics 19(3), 1614–1625.

18. R Development Core Team (2006), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. Internet. http://www.R-project.org

19. Smith, K. (1918), ‘On the Standard Deviations of Adjusted and Interpolates Values of an Observed Polynomial Functions and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of Observations’, Biometrika 12, 1–85.

How to Cite

APA

López, V. I. and Ramos, R. (2007). UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS. Revista Colombiana de Estadística, 30(1), 37–51. https://revistas.unal.edu.co/index.php/estad/article/view/29317

ACM

[1]
López, V.I. and Ramos, R. 2007. UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS. Revista Colombiana de Estadística. 30, 1 (Jan. 2007), 37–51.

ACS

(1)
López, V. I.; Ramos, R. UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS. Rev. colomb. estad. 2007, 30, 37-51.

ABNT

LÓPEZ, V. I.; RAMOS, R. UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS. Revista Colombiana de Estadística, [S. l.], v. 30, n. 1, p. 37–51, 2007. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/29317. Acesso em: 4 aug. 2024.

Chicago

López, Víctor Ignacio, and Rogelio Ramos. 2007. “UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS”. Revista Colombiana De Estadística 30 (1):37-51. https://revistas.unal.edu.co/index.php/estad/article/view/29317.

Harvard

López, V. I. and Ramos, R. (2007) “UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS”, Revista Colombiana de Estadística, 30(1), pp. 37–51. Available at: https://revistas.unal.edu.co/index.php/estad/article/view/29317 (Accessed: 4 August 2024).

IEEE

[1]
V. I. López and R. Ramos, “UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS”, Rev. colomb. estad., vol. 30, no. 1, pp. 37–51, Jan. 2007.

MLA

López, V. I., and R. Ramos. “UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS”. Revista Colombiana de Estadística, vol. 30, no. 1, Jan. 2007, pp. 37-51, https://revistas.unal.edu.co/index.php/estad/article/view/29317.

Turabian

López, Víctor Ignacio, and Rogelio Ramos. “UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS”. Revista Colombiana de Estadística 30, no. 1 (January 1, 2007): 37–51. Accessed August 4, 2024. https://revistas.unal.edu.co/index.php/estad/article/view/29317.

Vancouver

1.
López VI, Ramos R. UNA INTRODUCCIÓN A LOS DISEÑOS ÓPTIMOS. Rev. colomb. estad. [Internet]. 2007 Jan. 1 [cited 2024 Aug. 4];30(1):37-51. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/29317

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