Universal optimality of digital nets and lattice designs. (English) Zbl 1178.62080
Summary: This article considers universal optimality of digital nets and lattice designs in a regression model. Based on the equivalence theorem for matrix means and majorization theory, necessary and sufficient conditions for lattice designs being \(\varphi _{p }\)- and universally optimal in trigonometric function and Chebyshev polynomial regression models are obtained. It is shown that digital nets are universally optimal for both complete and incomplete Walsh function regression models under some specified conditions, and are also universally optimal for complete Haar wavelet regression models but may not for incomplete Haar wavelet regression models.
MSC:
62K05 | Optimal statistical designs |
62K15 | Factorial statistical designs |
62G08 | Nonparametric regression and quantile regression |
62K10 | Statistical block designs |
References:
[1] | Pukelsheim F. Optimal Design of Experiments. New York: Wiley, 1993 · Zbl 0834.62068 |
[2] | Bate R A, Buck R J, Riccomagno E, et al. Experimental design and observation for large systems. J Roy Statist Soc Ser B, 58: 77–94 (1996) · Zbl 0850.62627 |
[3] | Kiefer J, Wolfowitz J. Optimum designs in regression problems. Ann Math Statist, 30: 271–294 (1959) · Zbl 0090.11404 · doi:10.1214/aoms/1177706252 |
[4] | Riccomagno E, Schwabe R, Wynn H P. Lattice-based D-optimum design for Fourier regression. Ann Statist, 25: 2313–2327 (1997) · Zbl 0895.62081 · doi:10.1214/aos/1030741074 |
[5] | Arnold B C. Majorization and the Lorenz Order: A Brief Introduction. Berlin: Springer-Verlag, 1987 · Zbl 0649.62041 |
[6] | Marshall A W, Olkin I. Inequalities: Theory of Majorization and Its Applications. New York: Academic Press, 1979 · Zbl 0437.26007 |
[7] | Bondar J V. Universal optimality of experimental designs: definitions and a criterion. Canad J Statist, 11: 325–331 (1983) · Zbl 0539.62087 · doi:10.2307/3314890 |
[8] | Mason J C, Handscomb D C. Chebyshev Polynomials. Florida: Chapman & Hall/CRC, 2003 |
[9] | Kiefer J, Studden W J. Optimal designs for large degree polynomial regression. Ann Statist, 4: 1113–1123 (1976) · Zbl 0357.62051 · doi:10.1214/aos/1176343646 |
[10] | Lim Y B, Studden W J. Efficient D s-optimal designs for multivariate polynomial regression on the q-cube. Ann Statist, 16: 1225–1240 (1988) · Zbl 0664.62075 · doi:10.1214/aos/1176350957 |
[11] | Niederreiter H. Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM, 1992 · Zbl 0761.65002 |
[12] | Hickernell F J, Dick J. An algorithm-driven approach to error analysis for multidimensional integration. Int J Numer Anal Model, 5: 167–189 (2008) · Zbl 1131.41316 |
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