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An extension of a central limit theorem of F. EICKER for linear vector forms is used to derive conditions for the asymptotic normality of Least-Square Estimators in multivariate linear singular models with stochastically independent error terms. These conditions are shown to be necessary too in some cases. Several previous results concerning asymptotic normality of Least-Square Estimators in regular models are special cases of the theory developed in this paper.

The conditions guarantying consistency and asymptotic normality of Least-Square estimators as well as the consistency of the usual tests for linear hypotheses in multivariate linear models are shown to be valid whenever almost optimal experimental designs are used.

Usually the variance of independent observations resulting from a linear or a nonlinear relationship is estimated by the Least-Squares residual estimator. In this paper its asymptotic properties are investigated. Further the asymptotic behaviour of tests for one-sided hypotheses on the variance is studied. The paper splits into two parts, the first one concerned with linear and the second one with nonlinear models.

The general nonlinear regression model is considered, where the error distribution can be nonnormal. In this paper we present tests, confidence intervals and estimators for both the regression coefficients and the variance of the observations which perform well in the sense of second order asymptotic theory. In particular, robustness results for deviations from the normal distribution are given including distributions with either known or unknown skewness and kurtosis.