In this article a new monotone nonparametric estimate for a regression function of two or more variables is proposed. The method starts with an unconstrained nonparametric regression estimate and uses successively one-dimensional isotonization procedures. In the case of a strictly monotone regression function, it is shown that the new estimate is first order asymptotic equivalent to the unconstrained estimate, and asymptotic normality of an appropriate standardization of the estimate is established. Moreover, if the regression function is not monotone in one of its arguments, the constructed estimate has approximately the same L p -norm as the initial unconstrained estimate. The methodology is also illustrated by means of a simulation study, and two data examples are analyzed.

To estimate the effective dose level ED α in the common binary response model, several parametric and nonparametric estimators have been proposed in the literature. In the present paper, we focus on nonparametric methods and present a detailed numerical comparison of four different approaches to estimate the ED α nonparametrically. The methods are briefly reviewed and their finite sample properties are studied by means of a detailed simulation study. Moreover, a data example is presented to illustrate the different concepts.

Conditional quantiles, Additive models, Marginal integration, Non-increasing rearrangements,