Sequential Construction of an Experimental Design from an I.I.D. Sequence of Experiments without Replacement

Abstract

We consider a regression problem, with observations y _k = η(θ, ξ _k ) + ϵ _k , where {ϵ _k } is an i.i.d. sequence of measurement errors and where the experimental conditions £& form an i.i.d. sequence of random variables, independent of {ϵ _k }, which are observed sequentially. The length of the sequence {ξ _k } is N but only n < N experiments can be performed. As soon as a new experiment ξ _k is available, one must decide whether to perform it or not. The problem is to choose the n values ${\xi _{{k_1}}},.....{\xi _{{k_n}}}$ at which observations ${y_{{k_1}}},.....,{y_{{k_n}}}$ will be made in order to estimate the parameters θ. An optimal rule for the on-line selection of (math) is easily determined when p = dim θ = 1. A suboptimal open-loop feedback-optimal rule is suggested in Pronzato (1999b) for the case p > 1. We propose here a different suboptimal solution, based on a one-step-ahead optimal approach. A simple procedure, derived from an adaptive rule which is asymptotically optimal, Pronzato (1999a), when p = 1 (N → ∞, n fixed), is presented. The performances of these different strategies are compared on a simple example.