A Kiefer-Wolfowitz comparison theorem for wicksell's problem

We extend the isotonic analysis for Wicksell's problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer-Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate n(-1) log n faster than n(-2/3) log n. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and smooth estimators.