## The Annals of Statistics

### Complete classes of designs for nonlinear regression models and principal representations of moment spaces

#### Abstract

In a recent paper Yang and Stufken [Ann. Statist. 40 (2012a) 1665–1685] gave sufficient conditions for complete classes of designs for nonlinear regression models. In this note we demonstrate that there is an alternative way to validate this result. Our main argument utilizes the fact that boundary points of moment spaces generated by Chebyshev systems possess unique representations.

#### Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1260-1267.

Dates
First available in Project Euclid: 13 June 2013

https://projecteuclid.org/euclid.aos/1371150900

Digital Object Identifier
doi:10.1214/13-AOS1108

Mathematical Reviews number (MathSciNet)
MR3113810

Zentralblatt MATH identifier
1293.62156

Subjects
Primary: 62K05: Optimal designs

#### Citation

Dette, Holger; Schorning, Kirsten. Complete classes of designs for nonlinear regression models and principal representations of moment spaces. Ann. Statist. 41 (2013), no. 3, 1260--1267. doi:10.1214/13-AOS1108. https://projecteuclid.org/euclid.aos/1371150900

#### References

• Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: A review. Statist. Sci. 10 273–304.
• Chernoff, H. (1953). Locally optimal designs for estimating parameters. Ann. Math. Statist. 24 586–602.
• Dette, H. (1997). Designing experiments with respect to “standardized” optimality criteria. J. R. Stat. Soc. Ser. B Stat. Methodol. 59 97–110.
• Dette, H., Melas, V. B. and Wong, W. K. (2006). Locally $D$-optimal designs for exponential regression models. Statist. Sinica 16 789–803.
• Dette, H. and Melas, V. B. (2011). A note on the de la Garza phenomenon for locally optimal designs. Ann. Statist. 39 1266–1281.
• Fang, X. and Hedayat, A. S. (2008). Locally $D$-optimal designs based on a class of composed models resulted from blending $E_{\max}$ and one-compartment models. Ann. Statist. 36 428–444.
• He, Z., Studden, W. J. and Sun, D. (1996). Optimal designs for rational models. Ann. Statist. 24 2128–2147.
• Jennrich, R. I. (1969). Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40 633–643.
• Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Pure and Applied Mathematics XV. Wiley, New York.
• Khuri, A. I., Mukherjee, B., Sinha, B. K. and Ghosh, M. (2006). Design issues for generalized linear models: A review. Statist. Sci. 21 376–399.
• Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849–879.
• Pukelsheim, F. (2006). Optimal Design of Experiments. Classics in Applied Mathematics 50. SIAM, Philadelphia, PA.
• Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models. Dekker, New York.
• Seber, G. A. F. and Wild, C. J. (1989). Nonlinear Regression. Wiley, New York.
• Yang, M. (2010). On the de la Garza phenomenon. Ann. Statist. 38 2499–2524.
• Yang, M. and Stufken, J. (2009). Support points of locally optimal designs for nonlinear models with two parameters. Ann. Statist. 37 518–541.
• Yang, M. and Stufken, J. (2012a). Identifying locally optimal designs for nonlinear models: A simple extension with profound consequences. Ann. Statist. 40 1665–1685.
• Yang, M. and Stufken, J. (2012b). On locally optimal designs for generalized linear models with group effects. Statist. Sinica 22 1765–1786.