Asymptotic Normality of Nonlinear Least Squares under Singular Experimental Designs

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).