For each r ∈ N we prove the Nikolskii type pointwise estimate for coconvex approximation of functions f ∈ W , the subspace of all functions f ∈ C[−1, 1], possessing an absolutely continuous (r−1)st derivative on (−1, 1) and satisfying f (r) ∈ L∞[−1, 1], that change their convexity once on [−1, 1].

The asymptotic distribution of sample quantiles in the classical definition is well-known to be normal for absolutely continuous distributions. However, this is no longer true for discrete distributions or sampleswith ties.We show that the definition of sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample q...

The constructive definition usually begins with a function f, then by the process of using Riemann sums and limits, we arrive the definition of the integral of f, ∫ b a f. On the other hand, a descriptive definition starts with a primitive F satisfying certain condition(s) such as F ′ = f and F is absolutely continuous if f is Lebesgue integrable, and F is generalized absolutely continuous if f...