Probability distribution of the multivariate nonlinear least squares estimates

The nonlinear regression model yi = f}t(0u . . . . 0,„) + et with (e1, . . . , eN) ~ A (0, J)) a r , d with //,(.) twice continuously differentiable is considered. Under the assumption that the maximal curvature of the mean-values manifold {ii(0) :0eU}cR is bounded, an approximative probability density for the least squares estimates of (0U . . . , 6m) is proposed. This density depends on the first form ( = the information matrix) and on the second form of the mean-values manifold (Eq. (9)). The level of approximation depends on the probability that the sample goes beyond the nearest center of curvature of the mean-values manifold and it is expressed in the paper (Theorem 1).