Concordance-based predictive measures in regression models for discrete responses

Dependence measures are often used in practice in order to assess the quality of a regression model. This is for instance the case with Kendall’s tau and other association coefficients based on concordance probabilities. However, in case the response variable is discrete, correlation indices are often bounded and restricted to a sub-interval of [−1, 1]. Hence, in this context, small positive values of Kendall’s tau may actually support goodness of prediction when getting close to its highest attainable value. In this paper, we derive the best-possible upper bounds for Kendall’s tau when the response variable is discrete. Two cases are considered, depending on whether the score is continuous or discrete. Also, we illustrate the obtained upper bounds on a motor third-party liability insurance portfolio.