Optimum Design 2000 pp 113-122 | Cite as

# 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

*. An optimal rule for the on-line selection of (math) is easily determined when*

**θ***p*= dim

*= 1. A suboptimal open-loop feedback-optimal rule is suggested in Pronzato (1999b) for the case*

**θ***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.

## Keywords

Sequential design random experiments expected determinant## Preview

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