An algorithm for monotone regression with one or more independent variables

SUMMARY Two related algorithms are given to find a monotone function of one or more variables that best approximates in the least squares sense given function values that are not already monotone. For one independent variable, such algorithms are well known; see, for example, Bartholomew (1959). This case is briefly repeated in ? 2 for completeness and as an introduction to the more difficult case of two or more independent variables. Of the two algorithms given in ?? 3 and 5, the second one should generally be shorter. Section 4 contains an auxiliary algorithm that is needed in connexion with both ?? 3 and 5. The paper closes with an example in ? 6 and some remarks in ? 7.