Asymptotic normality of nonlinear least squares under singular experimental designs
Résumé
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) +epsilon _i, i=1,2... and independent errors epsilon_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... to the limiting singular design measure xi, the convergence of the estimator of h(theta) 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 xi (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 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).
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