Least squares estimates of regression functions with certain monotonicity and concavity/convexity restrictions

Abstract

In all regression problems the choice of model and estimation method is due to a priori information about the regression function. In some situations it is motivated to consider regression functions with specific non-parametric characteristics, for instance monotonicity and/or concavity/convexity. We propose a new least squares estimation method for curves that fulfil monotonicity and concavity/convexity restrictions. The least squares estimate of such a regression function is a piecewise linear continuous function with bending points contained in the set of the observed values of the independent variable. The set of bending points, which makes the function a least squares solution can be determined by an iterative algorithm within a finite number of steps.