Non-linear regression with multidimensional indices

We consider a non-linear regression model when the index variable is multidimensional. Sufficient conditions on the non-linear function are given under which the least-squares estimators are strongly consistent and asymptotically normally distributed. These sufficient conditions are satisfied by harmonic type functions, which are also of interest in the one dimensional index case where Wu's (Asymptotic theory of non-linear least-squares estimation, Ann. Statist. 9 (1981) 501-513) and Jennrich's (Asymptotic properties of non-linear least-squares estimators, Ann. Math. Statist. 40 (1969) 633-643) sufficient conditions are not applicable.