Constrained \(D\)- and \(D_ 1\)-optimal designs for polynomial regression. (English) Zbl 1105.62360

Dette, Holger; Franke, Tobias

Ann. Stat. 28, No. 6, 1702-1727 (2000).

Summary: In the common polynomial regression model of degree m we consider the problem of determining the \(D\)- and \(D_1\)-optimal designs subject to certain constraints for the \(D_1\)-efficiencies in the models of degree \(m-j\), \(m-j+1,\dots, m + k\;(m > j\geq 0, k\geq 0 \text{ given})\). We present a complete solution of these problems, which on the one hand allow a fast computation of the constrained optimal designs and, on the other hand, give an answer to the question of the existence of a design satisfying all constraints. Our approach is based on a combination of general equivalence theory with the theory of canonical moments. In the case of equal bounds for the \(D_1\)-efficiencies the constrained optimal designs can be found explicitly by an application of recent results for associated orthogonal polynomials.

MSC:

62K05	Optimal statistical designs
62J02	General nonlinear regression

References:

[1]	Box, G. E. P. and Draper, N. R. (1959). A basis for the selection of a response surface design. J. Amer. Statist. Assoc. 54 622-654. JSTOR: · Zbl 0116.36804 · doi:10.2307/2282542
[2]	Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. · Zbl 0389.33008
[3]	Clyde, M. and K. Chaloner (1996). The equivalence of constrained and weighted designs in multiple objective design problems. J. Amer. Statist. Assoc. 91 1236-1244. JSTOR: · Zbl 0883.62079 · doi:10.2307/2291742
[4]	Cook, R. D. and Nachtsheim, C. (1982). Model-robust, linear-optimal design. Technometrics 24 49-55. JSTOR: · Zbl 0483.62063 · doi:10.2307/1267577
[5]	Cook, R. D. and Wong, W. K. (1994). On the equivalence of constrained and compound optimal designs. J. Amer. Statist. Assoc. 89 687-692. JSTOR: · Zbl 0799.62081 · doi:10.2307/2290872
[6]	Dette, H. (1990). A generalization of Dand D1-optimal designs in polynomial regression. Ann. Statist. 18 1784-1805. Dette, H. (1995a). Discussion of ”Constrained optimization of experimental design,” by V. Fedorov and D. Cook. Statistics 26 153-161. Dette, H. (1995b). Optimal designs for identifying the degree of a polynomial regression. Ann. Statist. 23 1248-1266. · Zbl 0714.62068
[7]	Dette, H. and Studden, W. J. (1997). The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis. Wiley, New York. · Zbl 0886.62002
[8]	Franke, T. (2000). Dund D1-optimale Versuchspläne unter Nebenbedingungen und gewichtete Maximin-Versuchspläne bei polynomialer Regression. Dissertation, Ruhr-Univ., Bochum (in German). · Zbl 0998.62514
[9]	Grosjean, C. C. (1986). The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials. J. Comput. Appl. Math. 18 259-307. · Zbl 0627.33008 · doi:10.1016/0377-0427(86)90001-4
[10]	Guest, P. G., (1958). The spacing of observations in polynomial regression. Ann. Math. Statist. 29 294-299. · Zbl 0087.15303 · doi:10.1214/aoms/1177706730
[11]	Hoel, P. G. (1958). Efficiency problems in polynomial estimation. Ann. Math. Statist. 29 1134-1145. · Zbl 0094.14501 · doi:10.1214/aoms/1177706446
[12]	Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849-879. · Zbl 0291.62093 · doi:10.1214/aos/1176342810
[13]	Kiefer, J. C. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271-294. · Zbl 0090.11404 · doi:10.1214/aoms/1177706252
[14]	Lasser, E. (1994). Orthogonal polynomials and hypergroups II: the symmetric case. Trans. Amer. Math. Soc. 341 749-770. · Zbl 0804.42013 · doi:10.2307/2154581
[15]	Lau, T. S. (1983). Theory of canonical moments and its applications in polynomial regression I, II. Technical reports 83-23, 83-24. Purdue Univ.
[16]	Lau, T. S. (1988). D-optimal designs on the unit q-ball. J. Statist. Plann. Inf. 19 299-315. · Zbl 0850.62603 · doi:10.1016/0378-3758(88)90038-9
[17]	Läuter, E. (1974). Experimental design in a class of models. Math. Oper. Statist. 5 673-708. Lee, C. M. S. (1988a). Constrained optimal designs. J. Statist. Plann. Inf. 18 377-389. Lee, C. M. S. (1988b). D-optimal designs for polynomial regression, when lower degree parameters are more important. Utilitas Math. 34 53-63.
[18]	Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. · Zbl 0834.62068
[19]	Skibinsky, M. (1967). The range of the n + 1 th moment for distributions on 0 1. J. Appl. Probab. 4 543-552. JSTOR: · Zbl 0189.18803 · doi:10.2307/3212220
[20]	Skibinsky, M. (1986). Principal representations and canonical moment sequences for distributions on an interval. J. Math. Anal. Appl. 120 95-120. · Zbl 0607.60013 · doi:10.1016/0022-247X(86)90207-6
[21]	Stigler, S. (1971). Optimal experimental designs for polynomial regression. J. Amer. Statist. Assoc. 66 311-318. · Zbl 0217.51701 · doi:10.2307/2283928
[22]	Studden, W. J. (1980). Ds-optimal designs for polynomial regression using continued fractions. Ann. Statist. 8 1132-1141. Studden, W. J. (1982a). Optimal designs for weighted polynomial regression using canonical moments. In Third Purdue Symposium on Decision Theory and Related Topics 2 (S. S. Gupta and J. O. Berger, eds.) 335-350. Academic Press, New York. Studden, W. J. (1982b). Some robust type D-optimal designs in polynomial regression. J. Amer. Statist. Assoc. 8 916-921. · Zbl 0447.62070 · doi:10.1214/aos/1176345150
[23]	Studden, W. J. (1989). Note on some p-optimal designs for polynomial regression. Ann. Statist. 17 618-623. · Zbl 0675.62047 · doi:10.1214/aos/1176347129
[24]	Szeg ö, G. (1975). Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. 23. · Zbl 0305.42011