Isotonic smoothing splines under sequential designs

Isotonic smoothing splines are introduced as a natural extension of ordinary isotonic estimates in the estimation of a regression function p(x)=E{ YIX=x}. A constructive characterization for the isotonic smoothing splines is given. Conditions are given for consistency under sequential designs, where the observation points are random and the experimenter after observing X, may choose to observe or to skip observing Yx.