Comparing the shapes of regression functions

Does a regression function follow a specified shape? Do two regression functions have the same shape? How can regression functions be grouped, based on shape? These questions can occur when investigating monotonicity, when counting local maxima or when studying variation in families of curves. One can address these questions by considering the rank correlation coefficient between two functions. This correlation is a generalisation of the rank correlation between two finite sets of numbers and is equal to one if and only if the two functions have the same shape. A sample rank correlation based on smoothed estimates of the regression functions consistently estimates the true correlation This sample rank correlation can be used as a measure of similarity between functions in cluster analysis and as a measure of monotonicity or modality.