Abstract
In order to study the asymptotic properties of estimators, we need to indicate how the sequence of design points x 1, x 2, … in \(\mathcal{X} \subset {\mathbb{R}}^{d}\) is generated, i.e., specify some properties of the experimental design.
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Notes
- 1.
A subset of a metric space is totally bounded if it can be covered by a finite number of balls with radius ε for any ε, which is weaker than compactness. A subset of an Euclidean space Θ is totally bounded if and only if it is bounded, which compared to compactness relaxes the condition of Θ being closed. In the next chapters, the assumption of compactness of Θ will force us to work in the interior of Θ when differentiability will be required, in particular, to obtain the asymptotic normality of estimators.
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Pronzato, L., Pázman, A. (2013). Asymptotic Designs and Uniform Convergence. In: Design of Experiments in Nonlinear Models. Lecture Notes in Statistics, vol 212. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6363-4_2
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