×

zbMATH — the first resource for mathematics

Robust designs for polynomial regression by maximizing a minimum of \(D\)- and \(D_1\)-efficiencies. (English) Zbl 1012.62080
Summary: In the common polynomial regression of degree m we determine the design which maximizes the minimum of the \(D\)-efficiency in the model of degree m and the \(D_1\)-efficiencies in the models of degree \(m-j,\dots, m +k\) \((j, k\geq 0\) given). The resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness-of-fit of the assumed model of degree \(m\) by testing the highest coefficients in the polynomials of degree \(m-j,\dots, m +k\).
Our approach is based on a combination of the theory of canonical moments and general equivalence theory for minimax optimality criteria. The optimal designs can be explicitly characterized by evaluating certain associated orthogonal polynomials.

MSC:
62K05 Optimal statistical designs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. Dover, New York. · Zbl 0171.38503
[2] 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
[3] Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. · Zbl 0389.33008
[4] Dette, H. (1990). A generalization of Dand D1-optimal design in polynomial regression. Ann. Statist. 18 1784-1804. · Zbl 0714.62068
[5] Dette, H. (1995). Optimal designs for identifying the degree of a polynomial regression. Ann. Statist. 23 1248-1266. · Zbl 0847.62064
[6] 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
[7] Franke, T. (2000). Du nd D1-optimale Versuchspläne unter Nebenbedingungen und gewichtete Maximin-Versuchspläne bei polynomialer Regression. Dissertation, RuhrUniv. Bochum (in German). · Zbl 0998.62514
[8] 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. Comp. Appl. Math. 18 259-307. · Zbl 0627.33008
[9] Guest, P. G. (1958). The spacing of observations in polynomial regression. Ann. Math. Statist. 29 294-299. · Zbl 0087.15303
[10] Hoel, P. G. (1958). Efficiency problems in polynomial estimation. Ann. Math. Statist. 29 1134-1145. · Zbl 0094.14501
[11] Huber, P. J. (1975). Robustness and designs. In A Survey of Statistical Designs (J. N. Srivastava, ed.) 287-301. North Holland, Amsterdam. · Zbl 0318.62035
[12] Kiefer, J. C. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849-879. · Zbl 0291.62093
[13] Kiefer, J. C. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271-294. · Zbl 0090.11404
[14] Lasser, R. (1994). Orthogonal polynomials and hypergroups II: the symmetric case. Trans. Amer. Math. Soc. 341 749-770. · Zbl 0804.42013
[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. Inference 19 299-315. · Zbl 0850.62603
[17] Läuter, E. (1974). Experimental design in a class of models. Math. Oper. Statist. 5 379-398. · Zbl 0297.62056
[18] Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. · Zbl 0834.62068
[19] Pukelsheim, F. and Studden, W. J. (1993). E-optimal designs for polynomial regression. Ann. Statist. 21 402-415. · Zbl 0787.62075
[20] Silvey, S. D. (1980). Optimal Design. Chapman and Hall, London. · Zbl 0468.62070
[21] 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
[22] Skibinsky, M. (1986). Principal representations and canonical moment sequences for distributions on an interval. J. Math. Anal. Appl. 120 95-120. · Zbl 0607.60013
[23] Spruill, M. G. (1990). Good designs for testing the degree of a polynomial mean. Sankhy\?a Ser. B 52 67-74. · Zbl 0733.62075
[24] Stigler, S. (1971). Optimal experimental design for polynomial regression. J. Amer. Statist. Assoc. 66 311-318. · Zbl 0217.51701
[25] Studden, W. J. (1968). Optimal designs on Tchebycheff points. Ann. Math. Statist. 39 1435-1447. · Zbl 0174.22404
[26] 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 Statistical Decision Theory and Related Topics III 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. 77 916-921. · Zbl 0447.62070
[27] Studden, W. J. (1989). Note on some p-optimal designs for polynomial regression. Ann. Statist. 17 618-623. · Zbl 0675.62047
[28] Szeg ö, G. (1975). Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ., Providence, RI. · Zbl 0305.42011
[29] Wiens, D. P. (1992). Minimax designs for approximately linear regression. J. Statist. Plann. Inference 31 353-371. · Zbl 0755.62058
[30] Wong, W. K. (1994). Comparing robust properties of A-, D-, Eand G-optimal designs. Comput. Statist. Data Anal. 18 441-448. · Zbl 0825.62646
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.