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Optimal designs for a class of nonlinear regression models. (English) Zbl 1056.62084

Summary: For a broad class of nonlinear regression models we investigate the local \(E\)- and \(c\)-optimal design problem. It is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the Chebyshev points, which are the local extrema of the equi-oscillating best approximation of the function \(f_0\equiv 0\) by a normalized linear combination of the regression functions in the corresponding linearized model. The class of models includes rational, logistic and exponential models and for the rational regression models the \(E\)- and \(c\)-optimal design problem is solved explicitly in many cases.

MSC:

62K05 Optimal statistical designs
62J02 General nonlinear regression
41A50 Best approximation, Chebyshev systems
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[1] Becka, M., Bolt, H. M. and Urfer, W. (1993). Statistical evalutation of toxicokinetic data. Environmetrics 4 311–322.
[2] Becka, M. and Urfer, W. (1996). Statistical aspects of inhalation toxicokinetics. Environ. Ecol. Stat. 3 51–64.
[3] Chaloner, K. and Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. J. Statist. Plann. Inference 21 191–208. · Zbl 0666.62073 · doi:10.1016/0378-3758(89)90004-9
[4] Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: A review. Statist. Sci. 10 273–304. JSTOR: · Zbl 0955.62617 · doi:10.1214/ss/1177009939
[5] Chernoff, H. (1953). Locally optimal designs for estimating parameters. Ann. Math. Statist. 24 586–602. · Zbl 0053.10504 · doi:10.1214/aoms/1177728915
[6] Dette, H. and Haines, L. (1994). \(E\)-optimal designs for linear and nonlinear models with two parameters. Biometrika 81 739–754. · Zbl 0816.62055 · doi:10.1093/biomet/81.4.739
[7] Dette, H., Haines, L. and Imhof, L. A. (1999). Optimal designs for rational models and weighted polynomial regression. Ann. Statist. 27 1272–1293. · Zbl 0957.62062 · doi:10.1214/aos/1017938926
[8] Dette, H., Melas, V. B. and Pepelyshev, A. (2002). Optimal designs for a class of nonlinear regression models. Preprint, Ruhr-Universität Bochum. Available at www.ruhr-uni-bochum.de/mathematik3/preprint.htm. · Zbl 1056.62084 · doi:10.1214/009053604000000382
[9] Dette, H., Melas, V. B. and Pepelyshev, A. (2004). Optimal designs for estimating individual coefficients in polynomial regression—a functional approach. J. Statist. Plann. Inference 118 201–219. · Zbl 1031.62055 · doi:10.1016/S0378-3758(02)00397-X
[10] Dette, H. and Studden, W. J. (1993). Geometry of \(E\)-optimality. Ann. Statist. 21 416–433. JSTOR: · Zbl 0780.62057 · doi:10.1214/aos/1176349034
[11] Dette, H. and Wong, W. K. (1999). \(E\)-optimal designs for the Michaelis–Menten model. Statist. Probab. Lett. 44 405–408. · Zbl 0940.62066 · doi:10.1016/S0167-7152(99)00033-4
[12] DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, New York. · Zbl 0797.41016
[13] Dudzinski, M. L. and Mykytowycz, R. (1961). The eye lens as an indicator of age in the wild rabbit in Australia. CSIRO Wildlife Research 6 156–159.
[14] Elfving, G. (1952). Optimum allocation in linear regression theory. Ann. Math. Statist. 23 255–262. · Zbl 0047.13403 · doi:10.1214/aoms/1177729442
[15] Ford, I. and Silvey, S. D. (1980). A sequentially constructed design for estimating a nonlinear parametric function. Biometrika 67 381–388. · Zbl 0433.62054 · doi:10.1093/biomet/67.2.381
[16] Ford, I., Torsney, B. and Wu, C.-F. J. (1992). The use of a canonical form in the construction of locally optimal designs for non-linear problems. J. Roy. Statist. Soc. Ser. B 54 569–583. · Zbl 0774.62080
[17] He, Z., Studden, W. J. and Sun, D. (1996). Optimal designs for rational models. Ann. Statist. 24 2128–2147. · Zbl 0867.62063 · doi:10.1214/aos/1069362314
[18] Heiligers, B. (1994). \(E\)-optimal designs in weighted polynomial regression. Ann. Statist. 22 917–929. JSTOR: · Zbl 0808.62065 · doi:10.1214/aos/1176325503
[19] Imhof, L. A. and Studden, W. J. (2001). \(E\)-optimal designs for rational models. Ann. Statist. 29 763–783. · Zbl 1012.62082 · doi:10.1214/aos/1009210689
[20] Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems : With Applications in Analysis and Statistics. Interscience, New York. · Zbl 0153.38902
[21] Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849–879. JSTOR: · Zbl 0291.62093 · doi:10.1214/aos/1176342810
[22] Kitsos, C. P., Titterington, D. M. and Torsney, B. (1988). An optimal design problem in rhythmometry. Biometrics 44 657–671. · Zbl 0707.62253 · doi:10.2307/2531581
[23] Melas, V. B. (1978). Optimal designs for exponential regression. Math. Operationsforsch. Statist. Ser. Statist. 9 45–59. · Zbl 0387.62062 · doi:10.1080/02331887808801407
[24] Melas, V. B. (2000). Analytical theory of \(E\)-optimal designs for polynomial regression. In Advances in Stochastic Simulation Methods (N. Balakrishan, V. B. Melas and S. Ermakov, eds.) 85–115. Birkhäuser, Boston. · Zbl 0998.62062
[25] Melas, V. B. (2001). Analytical properties of locally \(D\)-optimal designs for rational models. In MODA 6—Advances in Model-Oriented Design and Analysis (A. C. Atkinson, P. Hackel and W. G. Müller, eds.) 201–210. Physica, Heidelberg.
[26] Petrushev, P. P. and Popov, V. A. (1987). Rational Approximation of Real Functions . Cambridge Univ. Press. · Zbl 0644.41010
[27] Pronzato, L. and Walter, E. (1985). Robust experimental design via stochastic approximation. Math. Biosci. 75 103–120. · Zbl 0593.62070 · doi:10.1016/0025-5564(85)90068-9
[28] Pukelsheim, F. (1993). Optimal Design of Experiments . Wiley, New York. · Zbl 0834.62068
[29] Pukelsheim, F. and Studden, W. J. (1993). \(E\)-optimal designs for polynomial regression. Ann. Statist. 21 402–415. JSTOR: · Zbl 0787.62075 · doi:10.1214/aos/1176349033
[30] Pukelsheim, F. and Torsney, B. (1991). Optimal designs for experimental designs on linearly independent support points. Ann. Statist. 19 1614–1625. JSTOR: · Zbl 0729.62063 · doi:10.1214/aos/1176348265
[31] Ratkowsky, D. A. (1983). Nonlinear Regression Modeling : A Unified Practical Approach . Dekker, New York. · Zbl 0572.62054
[32] Ratkowsky, D. A. (1990). Handbook of Nonlinear Regression Models . Dekker, New York. · Zbl 0705.62060
[33] Seber, G. A. J. and Wild, C. J. (1989). Nonlinear Regression . Wiley, New York. · Zbl 0721.62062
[34] Silvey, S. D. (1980). Optimal Design . Chapman and Hall, London. · Zbl 0468.62070
[35] Studden, W. J. (1968). Optimal designs on Tchebycheff points. Ann. Math. Statist. 39 1435–1447. · Zbl 0174.22404
[36] Studden, W. J. and Tsay, J. Y. (1976). Remez’s procedure for finding optimal designs. Ann. Statist. 4 1271–1279. JSTOR: · Zbl 0347.62057 · doi:10.1214/aos/1176343659
[37] Szegö, G. (1975). Orthogonal Polynomials , 4th ed. Amer. Math. Soc., Providence, RI. · Zbl 0305.42011
[38] Wu, C.-F. J. (1985). Efficient sequential designs with binary data. J. Amer. Statist. Assoc. 80 974–984. · Zbl 0588.62133 · doi:10.2307/2288563
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