We consider Gaussian nonlinear regression models with constant information matrix ( = models with constant asymptotic variance) and models which are such after a repararnetrization (= “flat models”), including all one-dimensional nonlinear regression models. In is shown that a recently obtained nonasymptotical approximation of the probability density of the miximum likelihood (= least squares) estimator is particularly good in flat models. It is proved that under this approximative density the gradient of the squared distance between the true and the estimated means of the observed vector is nearly a normal random vector in models with constant information matrix. This allows to construct almost exact confidence regions in flat models, and to obtain approximative moments of the estimators