  In this paper, we consider binary response models with linear quantile
restrictions. Considerably generalizing previous research on this topic, our
analysis focuses on an infinite collection of quantile estimators. We derive a
uniform linearisation for the properly standardized empirical quantile process
and discover some surprising differences with the setting of continuously
observed responses. Moreover, we show that considering quantile processes
provides an effective way of estimating binary choice probabilities without
restrictive assumptions on the form of the link function, heteroskedasticity or
the need for high dimensional non-parametric smoothing necessary for approaches
available so far. A uniform linear representation and results on asymptotic
normality are provided, and the connection to rearrangements is discussed.
