Modelos para el propensity score que contemplan el supuesto de
positividad y su aplicación a problemas de datos faltantes y causalidad.

 Inverse probability weighted (IPW) estimators are widely used for estimating the average treatment effect in observational studies. The propensity score, defined as the conditional probability of treatment assignment given a set of covariates, is used in the construction of weights, which are necessary to achieve consistent estimators. IPW estimators are essentially weighted means of observed responses in which the weights are determined as the inverse of the estimated propensity score. In order to derive the asymptotic properties of IPW estimators, the propensity score is supposed to be bounded away from zero and one. This condition is known in the literature as positivity or overlap and, in practice, when it does not hold, IPW estimators are very unstable and have a large variability. In real data sets, a data generating process that violates the positivity assumption may lead to wrong inference because of the inaccuracy in the estimations. In this work, we attempt to conciliate between the positivity condition and the theory of generalized linear models by incorporating two extra parameters, which result in explicit bounds for the propensity score.