Optimal design for linear forms of the parameters in a Chebyshev regression

Michel Broniatowski (1) , Giorgio Celant (2)

1 LSTA - Laboratoire de Statistique Théorique et Appliquée
2 Department of Statistical Sciences

This paper of pedagogical nature considers optimal designs for linear combinations of the parameters in a Chebyshev regression scheme, namely defined by a vector c. Simple algebraic arguments lead to identify the class of such linear forms which admit unbiased linear estimators. This class is the Elfving set. Geometrical properties of this set provide a description of its frontier points as convex combinations of elements in the span of the regressors. The optimal design is shown to result from this representation. This statement is made precise in Elfving Theorem. However this Theorem does not provide an explicit form for the optimal design, but merely its existence and uniqueness. A further result due to Karlin and Studden provides a bridge between the optimal design properties and the theory of uniform approximation of functions by functions in a Chebyshev or Haar system of functions. This in turn provides the explicit form for the optimal design. The derivation of these results makes use of geometrical considerations pertaining to the class of moment matrices, following the approach by Pukelsheim.

Mots clés: Chebyshev regression; Elfving Theorem; moment matrix; optimal design; estimable function

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Michel Broniatowski, Giorgio Celant. Optimal design for linear forms of the parameters in a Chebyshev regression. Annales de l'ISUP, 2016, 60 (3), pp.15-54. ⟨hal-01010962v2⟩