Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs

A. Pázman () and L. Pronzato ()
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A. Pázman: Comenius University
L. Pronzato: Laboratoire I3S, CNRS - UNSA, Les Algorithmes – Bât. Euclide B

Chapter 8 in Optimal Design and Related Areas in Optimization and Statistics, 2009, pp 167-191 from Springer

Abstract: Summary We study the consistency and asymptotic normality of the LS estimator of a function h(θ) of the parameters θ in a nonlinear regression model with observations $$y_i=\eta(x_i,\theta) +\varepsilon_i$$ , $$i=1,2\ldots$$ and independent errors ε i . Optimum experimental design for the estimation of h(θ) frequently yields singular information matrices, which corresponds to the situation considered here. The difficulties caused by such singular designs are illustrated by a simple example: depending on the true value of the model parameters and on the type of convergence of the sequence of design points $$x_1,x_2\ldots$$ to the limiting singular design measure ξ, the convergence of the estimator of h(θ) may be slower than $$1/\sqrt{n}$$ , and, when convergence is at a rate of $$1/\sqrt{n}$$ and the estimator is asymptotically normal, its asymptotic variance may differ from that obtained for the limiting design ξ (which we call irregular asymptotic normality of the estimator). For that reason we focuss our attention on two types of design sequences: those that converge strongly to a discrete measure and those that correspond to sampling randomly from ξ. We then give assumptions on the limiting expectation surface of the model and on the estimated function h which, for the designs considered, are sufficient to ensure the regular asymptotic normality of the LS estimator of h(θ).

Keywords: Design Point; Asymptotic Normality; Asymptotic Variance; Design Measure; Design Sequence (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-0-387-79936-0_8

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DOI: 10.1007/978-0-387-79936-0_8

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