Median regression for ordered discrete response

We introduce a semiparametric median regression approach to ordered discrete responses, generalizing the maximum score estimation of Manski (1985) for binary response. The identification and a.s. consistency are proved. We assume that there exists a latent continuous variable, and several variations of the median regression approach depending on the restriction on the thresholds for the latent variable are examined. For some cases, bounds for the scale are derived. We also derive bounds for the finite sample breakdown point of the median regression method to show its robustness.