We represent new estimates of errors of quadrature formula, formula of numerical differentiation and approximation using Taylor polynomial. To measure the errors we apply representation of the remainder in Taylor formula by least concave majorant of the modulus of continuity of the n−th derivative of an n−times differentiable function. Our quantitative estimates are special applications of a mo...

We extend the isotonic analysis for the 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 usef...

We construct a continuous distribution G such that the number of faces in the smallest concave majorant of the random walk with G-distributed summands will take on each natural number infinitely often with probability one. This investigation is motivated by the fact that the number of faces Fn of the concave majorant of the random walk at time n has the same distribution as the number of record...