Simultaneous confidence bands for isotonic functions

Procedures for simultaneous confidence bands are provided for bivariate regression data {x,Yx,x∈X}{x,Yx,x∈X}, where EYxEYx is an unknown monotonic function of x. A new bandwidth procedure is introduced which generalizes the previously introduced procedures of Korn [1982. Confidence bands for isotonic dose–response curves. Appl. Statist. 31, 59–63] and Lee [1996. On estimation for monotone dose–response curves. J. Amer. Statist. Assoc. 91, 1110–1119]. To gain insight into the comparability and applicability of these procedures a unifying geometric framework is used. This framework leads to certain optimality results for these banding procedures and allows for extensions to settings that have not been previously considered.