An index of monotonicity and its estimation: a step beyond econometric applications of the Gini index

摘要

We suggest an index for measuring monotonicity of functions that can be used for determining which functions in a given class are more monotonic than others. Such results are of interest, for example, in reliability theory, survival analysis, and statistics where researchers want to know whether (unknown) failure rate, mean residual, density or other functions are monotonic or not, possibly only in some regions of their domains of definition. Therefore, we construct an empirical estimator for the aforementioned index of monotonicity and develop an appropriate asymptotic theory for it. Based on asymptotic results we then discuss testing hypotheses about monotonicity of (unknown) functions. The idea behind the construction of the index stems from geometric interpretations of the classical Lorenz curve and the Gini index, as well as from the far-reaching considerations by D. M. Cifarelli and E. Regazzini concerning general concentration functions and with them associated indices.