Monte Carlo and Quasi-Monte Carlo Methods 2000 pp 523-535 | Cite as
D-optimal Designs Based on Elementary Intervals for b-adic Haar Wavelet Regression Models
Abstract
Owen (1995) defined b-adic Haar wavelet in his study on Monte Carlo variance of scrambled net quadrature and showed that b-adic Haar wavelet is a useful tool in Monte Carlo and quasi-Monte Carlo methods. We in this paper study the problem of D-optimal design for both univariate and multivariate b-adic Haar wavelet models. A sufficient and necessary condition of D-optimal design for univariate b-adic Haar wavelet models is given. This result shows that the sufficient condition of Herzberg and Traves (1994) is also necessary. We find that D-optimal designs are also more general φp-optimal when b = 2 and indicate that this fact can not be extended to the case of b ≥ 3. Furthermore, one dimensional result is extended into multiple factor cases by the use of elementary intervals (Niederreiter, 1992). Finally, the D-optimality of orthogonal design under a multiple wavelet regression model is given.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Antoniadis, A. and Oppenheim, G. (1995), Wavelets and Statistics, Springer-Verlag, New York.MATHCrossRefGoogle Scholar
- 2.Bates, R. A., Buck, R. J., Riccomagno, R. and Wynn, H. P. (1996), Experimental design and observation for large systems, J. R. Statist. Soc. B, 58, 77–94.MathSciNetMATHGoogle Scholar
- 3.Cheng, C. S. (1995), Some projection properties of orthogonal designs, Ann. Statist., 23 1223–1233.MathSciNetMATHCrossRefGoogle Scholar
- 4.Chui, C. K. (1992), Wavelets: A Tutorial in Theory and Applications, Academic Press, Inc. Boston, San Diego.MATHGoogle Scholar
- 5.Daubechies, I. (1992), Ten Lectures on Wavelets, CBMS-NSF, SIAM, Philadelphia.MATHCrossRefGoogle Scholar
- 6.Fang, K. T. and Wang, Y. (1994), Number-theoretic Methods in Statistics, Chapman and Hall, London.Google Scholar
- 7.Herzberg, A. and Traves, W. H. (1994), An optimal experimental design for the Haar regression model, Canadian Journal of Statistics, 22, 357–364.MathSciNetMATHCrossRefGoogle Scholar
- 8.Kiefer, J. (1974), General equivalence theory for optimum designs approximate theory), Ann. Statist., 2, 849–879.MathSciNetMATHCrossRefGoogle Scholar
- 9.Niederreiter, H. (1992), Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NFS, SIAM, Philadelphia.MATHCrossRefGoogle Scholar
- 10.Owen, A. B. (1995), Monte Carlo Variance of Scrambled Net Quadrature. Proceeding's Workshop on Quasi-Monte Carlo Methods and Their Applications, eds. by Fang, K. T. and Hickernell, F. J., Hong Kong Baptist University, 69–102.Google Scholar
- 11.Pukelsheim, F.(1993), Optimal Design of Experiments, John Wiley & Sons, Inc.Google Scholar
- 12.Schwabe, R.(1994), Model robust experimental design in the presence of interactions: The orthogonal case. In PROBAS-TAT′94. Proc. Int. Conf. Math. Statist., Smolenice.Google Scholar
- 13.Schwabe, R.(1996), Optimum Designs for Multi-factor Models, Springer-Verlag New York, Inc.Google Scholar
- 14.Tang, B.(1993), Orthogonal design-based Latin hypercubes. Journal of Amer. Statist Assoc., 88, 1392–1397.Google Scholar