Assessing Monotonicity: An Approach Based on Transformed Order Statistics

In a number of research areas, such as non-convex optimization and machine learning, determining and assessing regions of monotonicity of functions is pivotal. Numerically, it can be done using the proportion of positive (or negative) increments of transformed ordered inputs. When the number of inputs grows, the proportion tends to an index of increase (or decrease) of the underlying function. In this paper, we introduce a most general index of monotonicity and provide its interpretation in all practically relevant scenarios, including those that arise when the distribution of inputs has jumps and flat regions, and when the function is only piecewise differentiable. This enables us to assess monotonicity of very general functions under particularly mild conditions on the inputs.