, Volume 65, Issue 3, pp 311–324 | Cite as

Estimation and optimal designs for linear Haar-wavelet models

  • Yongge TianEmail author
  • Agnes M. Herzberg


This paper gives an analytical expression for the best linear unbiased estimator (BLUE) of the unknown parameters in the linear Haar-wavelet model. From the analytical expression, we solve for the eigenvalues of the covariance matrix of the BLUE in analytical form. Further, we use these eigenvalues to construct some conventional discrete optimal designs for the model. The equivalences among these optimal designs are demonstrated and some examples are also given.


Haar-wavelet Linear model Best linear unbiased estimator Covariance matrix Information matrix Optimal design 

Mathematics Subject Classification



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  1. Antoniadis A, Grégoire G, McKeague IW (1994) Wavelet method for curve estimation. J Am Stat Assoc 89:1340–1342zbMATHCrossRefGoogle Scholar
  2. Fedorov VV (1972) Theory of optimal experiments. Translated from the Russian and edited by W. J. Studden and E. M. Klimko. Academic Press, New YorkGoogle Scholar
  3. Haar A (1910) Zur theorie der orthogonalen Funktionensysteme. Math Ann 69:331–371CrossRefMathSciNetGoogle Scholar
  4. Herzberg AM, Traves WN (1994) An optimal experimental design for the Haar regression model. Can J Stat 22:357–364zbMATHMathSciNetGoogle Scholar
  5. Oh H-S, Naveau P, Lee G (2001) Polynomial boundary treatment for wavelet regression. Biometrika 88:291–298zbMATHCrossRefMathSciNetGoogle Scholar
  6. Oyet AJ (2002) Minimax A- and D-optimal integer-valued wavelet designs for estimation. Can J Stat 30:301–316zbMATHMathSciNetGoogle Scholar
  7. Oyet AJ, Wiens DP (2000) Robust designs for wavelet approximations of regression models. J Nonparametric Stat 12:837–859zbMATHMathSciNetGoogle Scholar
  8. Rao GP (1983) Piecewise constant orthogonal functions and their application to systems and control. Springer, Berlin Heidelberg New YorkzbMATHCrossRefGoogle Scholar
  9. Xie M-Y (2002) D-optimal designs based on elementary intervals for b-adic Haar wavelet regression models. In: Niederreiter H, Spanier J (eds) Monte Carlo and quasi-Monte Carlo methods. Springer, Berlin Heidelberg New York, pp 523–535Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.School of EconomicsShanghai University of Finance and EconomicsShanghaiChina
  2. 2.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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