In this paper we derive inferential results for a new index of inequality, specifically defined for capturing significant changes observed both in the left and in the right tail of the income distributions. The latter shifts are an apparent fact for many countries like US, Germany, UK, and France in the last decades, and are a concern for many policy makers. We propose two empirical estimators for the index, and show that they are asymptotically equivalent. Afterwards, we adopt one estimator and prove its consistency and asymptotic normality. Finally we introduce an empirical estimator for its variance and provide conditions to show its convergence to the finite theoretical value. An analysis of real data on net income from the Bank of Italy Survey of Income and Wealth is also presented, on the base of the obtained inferential results.