The gradient allocation principle based on the higher moment risk measure

•Establish the consistency of the higher moment risk measure with stochastic dominance.•Prove that the higher moment risk measure is Gâteaux differentiable.•Derive an explicit expression for the associated capital allocation.•Establish the strong consistency and asymptotic normality of the empirical estimate.•Address the robustness issue via the influence function of the capital allocation.•Explore the interplay of the risk aversion and the confidence level involved.