Measuring the lack of monotonicity in functions
  Problems in econometrics, insurance, reliability engineering, and statistics quite often rely on the assumption that certain functions are non-decreasing. To satisfy this requirement, researchers frequently model the underlying phenomena using parametric and semi-parametric families of functions, thus effectively specifying the required shapes of the functions. To tackle these problems in a non-parametric way, in this paper we suggest indices for measuring the lack of monotonicity in functions. We investigate properties of the indices and also offer a convenient computational technique for practical use. 
Danang Teguh Qoyyimi and Ricardas Zitikis
stat.AP math.FA math.PR