This paper addresses important weaknesses in current methodology for the estimation of multivariate extreme event distributions. The estimation of the residual dependence index $η\in (0,1]$ is notoriously problematic. We introduce a flexible class of reduced-bias estimators for this parameter, designed to ameliorate the usual problems of threshold selection through a unified approach to familiar marginal standardisations. We derive the asymptotic properties of the proposed class of gradient estimators for $η$. Their efficiency stems from a hitherto neglected exponentially decaying term in the characterisation of the asymptotic independence based on the theory of regular variation. Simulation studies to demonstrate the finite-sample efficacy of the new gradient estimation across a wealth of bivariate distributions belonging to some max-domain of attraction that enjoy the asymptotic independence property. Our leading application illustrates how asymptotic independence can be discerned from monsoon-related rainfall occurrences at different locations in Ghana. The considerations involved in extending this framework to the estimation of the extreme value index attached to univariate domains of attraction associated with heavy-tailed distributions are briefly discussed.