The Asymptotic Behavior of Monotone Regression Estimates

Abstract

An estimator for a monotone regression function was proposed by Brunk. He has shown that if the underlying regression function has positive slope at a point, then, based on $r$ observations, the difference of the regression function and its estimate at that point has a nondegenerate limiting distribution if this difference is multiplied by $r^{1/3}$. To understand how the behavior of the regression function at a point influences the asymptotic properties of the estimator at that point, we have generalized Brunk's result to points at which the regression function does not have positive slope. If the first $\alpha - 1$ derivatives of the regression function are zero at a point and the $\alpha$th derivative is positive there, then the norming constants are of order $r^{\alpha/(2\alpha + 1)}$.