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We consider the estimation of nonlinear models with mismeasured explanatory variables, when information on the marginal distribution of the true values of these variables is available. We derive a semi-parametric MLE that is shown to be $\sqrt{n}$ consistent and asymptotically normally distributed. In a simulation experiment we find that the finite sample distribution of the estimator is close to the asymptotic approximation. The semi-parametric MLE is applied to a duration model for AFDC welfare spells with misreported welfare benefits. The marginal distribution of the correctly measured welfare benefits is obtained from an administrative source. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]