Explicit confidence bands and intervals for distribution functions and their derivatives via random Weierstrass-type operators

Abstract

Classical kernel estimators of second order are interpreted in terms of random Weierstrass-type operators, particularly random Steklov operators. This leads us to obtain explicit nonasymptotic confidence bands and intervals for distribution functions $F$ and their derivatives $F^{(k)}$. Under the only assumption that $F^{(k)}$ is uniformly continuous, confidence bands for $F^{(k)}$ are established by using the Dvoretzky-Kiefer-Wolfowitz inequality. To give confidence intervals, we allow $F^{(k)}$ to have isolated discontinuities of the first kind, so that we really estimate the midpoint function $(F^{(k)})_{\star}(x)$. The proofs are based either on concentration inequalities for subordinated stochastic processes or accurate estimates of the MSE of the corresponding estimators. The length of the confidence bands and intervals depends on the degree of smoothness of $F^{(k)}$ measured in terms of the second modulus of continuity. Both lengths are of order $n^{-1 / 2}$ if $F$ is locally a polynomial of degree $k+1$ at most.