ASYMPTOTIC NORMALITY OF NON-LINEAR LEAST SQUARES UNDER SINGULAR EXPERIMENTAL DESIGNS
Andrej Pazman and Luc Pronzato
(2005) Technical Report. Luc Pronzato, Sophia Antipolis, France.

## Abstract

We study the consistency and asymptotic normality of the LS estimator of a function $h(\theta)$ of the parameters $\theta$ in a nonlinear regression model with observations $y_i=\eta(x_i,\theta) +\varepsilon _i$, $i=1,2\ldots$ and independent errors $\mve_i$. Optimum experimental design for the estimation of $h(\theta)$ 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 $\xi$, the convergence of the estimator of $h(\theta)$ may be slower that $1/\sqrt{n}$, and, when convergence is in $1/\sqrt{n}$ and the estimator is asymptotically normal, its asymptotic variance may differ from that obtained for the limiting design $\xi$ (which we call {\em 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 $\xi$. 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 $h(\theta)$.