Simultaneous Choice of Design and Estimator in Nonlinear Regression with Parameterized Variance

In some nonlinear regression problems with parameterized variance both the design and the method of estimation have to be chosen. We compare asymptotically two methods of estimation: the penalized weighted LS (PWLS) estimator, which corresponds to maximum likelihood estimation (MLE) under the assumption of normal errors, and the two-stage LS (TSLS) estimator. We show that when the kurtosis κ of the distribution of the errors is zero, the asymptotic covariance matrix of the estimator is smaller for PWLS than for TSLS, which may not be the case when κ is not zero. We then suggest to construct two optimum designs, one for PWLS under the assumption κ = 0, the other for TSLS (with arbitrary κ), and compare their properties for different values of κ. All developments are made under the assumption of a randomized design, which allows rigorous proofs for the asymptotic properties of the estimators while avoiding the technical difficulties encountered in classical references such as Jennrich (1969) (finite tail product of the regression function and its derivatives, etc.).