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An optimal experimental design for the Haar regression model. (English) Zbl 0814.62039
This paper presents the general framework for optimal design problems. Wavelets were introduced, and the Haar wavelet system was given as an example. For the Haar regression model of order \(m\), any design which concentrates mass \(2^{-(m+1)}\) in the \(2^{m+1}\) intervals \[ \{[ 2^{-(m+1)} i, \;2^{-(m+1)} (i+1)) \}_{i= 0,1,\dots, 2^{m+1} -1} \] is D-optimal. Because the Haar wavelet family is a set of orthonormal functions, the vector of estimated regression parameters is particularly simple to calculate. If the optimal design is used for the Haar wavelet system, and if the number of observations used is equal to the number of terms in the regression, then the vector of estimated regression parameters is equivalent to the discrete wavelet transform of the data.
As wavelets become more used, one will want to compare the classical designs for ordinary linear regression with those of Fourier regression and wavelet regression. The use of the Haar wavelet is the first step. In some ways, the simplest wavelet, the Haar wavelet, has the advantage of its rectangular shape, which enables one to make better comparisons even by eye.

MSC:
62K05 Optimal statistical designs
33E15 Other wave functions
33E99 Other special functions
42C99 Nontrigonometric harmonic analysis
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