31b3b31a1c2f8a370206f111127c0dbd-Supplemental.pdf

Note that we allow multiple estimated quantiles to be identical to eachother,to accommodate the possibility of point masses. Furthermore, we assume ˆq0(x) and ˆq1(x) are conservative upper and lower bounds for the support ofY | X = x, i.e., ˆq0(X) = b0 < Y < bm = ˆq1(X). We will discuss in the next section practical options for estimating ˆq(x). Now, we leverage any givenˆq(x) to compute estimatesˆπj(x) of the unknown bin probabilities πj(x) in (6), for allj {1,...,m}. Although there are multiple way of doing this, a principled solution is to convert the information contained inˆq into a piece-wise constant density estimate, and then integrate that density within each bin.