Quantile and probability curves without crossing

Victor Chernozhukov

The most common approach to estimating conditional quantile curves is to fit a curve, typically linear, pointwise for each quantile. Linear functional forms, coupled with pointwise fitting, are used for a number of reasons including parsimony of the resulting approximations and good computational properties. The resulting fits, however, may not respect a logical monotonicity requirement - that the quantile curve be increasing as a function of probability. This paper studies the natural monotonization of these empirical curves induced by sampling from the estimated non-monotone model, and then taking the resulting conditional quantile curves that by construction are monotone in the probability. This construction of monotone quantile curves may be seen as a bootstrap and also as a monotonic rearrangement of the original non-monotone function. It is shown that the monotonized curves are...