  This paper considers endogenous selection models, in particular nonparametric
ones. Estimating the unconditional law of the outcomes is possible when one
uses instrumental variables. Using a selection equation which is additively
separable in a one dimensional unobservable has the sometimes undesirable
property of instrument monotonicity. We present models which allow for
nonmonotonicity and are based on nonparametric random coefficients indices. We
discuss their nonparametric identification and apply these results to inference
on nonlinear statistics such as the Gini index in surveys when the nonresponse
is not missing at random.
