Implicitly weighted robust regression estimators for linear and nonlinear regression models include linear and nonlinear versions of the least trimmed squares and least weighted squares. After recalling known facts about these estimators, a nonparametric bootstrap procedure is proposed in this paper for estimates of their variances. These bootstrap estimates are elaborated for both the linear and nonlinear model. Practical contributions include several examples investigating the performance of the nonlinear least weighted squares estimator and comparing it with the classical least squares also by means of the variance estimates. Another theoretical novelty is a proposal of a two-stage version of the nonlinear least weighted squares estimator with adaptive (data-dependent) weights.