A Rate of Convergence of a Distribution Connected with Integral Regression Function Estimation

Abstract

Brunk studied integral regression functions and has obtained strong laws and limiting distributions for estimators of these functions. In this note we will study additional conditions that ensure a rate of convergence of the distribution function of the maximum absolute difference of an integral regression function and its estimator, suitably normalized, to the distribution function of a normalized maximum absolute value of partial sums of random variables. These results are corollaries of convergence results obtained by Sawyer and Rosenkrantz.