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We propose various semiparametric estimators for nonlinear selection models, where slope and intercept can be separately identifed. When the selection equation satisfies a monotonic index restriction, we suggest a local polynomial estimator, using only observations for which the marginal distribution of instrument index is close to one. Such an estimator achieves a univariate nonparametric rate, which can range from a cubic to an 'almost' parametric rate. We then consider the case in which either the monotonic index restriction does not hold and/or the set of observations with propensity score close to one is thin so that convergence occurs at most at a cubic rate. We explore the finite sample behaviour in a Monte Carlo study, and illustrate the use of our estimator using a model for count data with multiplicative unobserved heterogeneity.