  Semiparametric discrete choice models are widely used in a variety of
practical applications. While these models are point identified in the presence
of continuous covariates, they can become partially identified when covariates
are discrete. In this paper we find that classical estimators, including the
maximum score estimator, (Manski (1975)), loose their attractive statistical
properties without point identification. First of all, they are not sharp with
the estimator converging to an outer region of the identified set, (Komarova
(2013)), and in many discrete designs it weakly converges to a random set.
Second, they are not robust, with their distribution limit discontinuously
changing with respect to the parameters of the model. We propose a novel class
of estimators based on the concept of a quantile of a random set, which we show
to be both sharp and robust. We demonstrate that our approach extends from
cross-sectional settings to classical static and dynamic discrete panel data
models.
