統計学輪講 第26回

This paper addresses the problem of risk aggregation with dependence uncertainty, where the marginal distributions of individual risks are known, but their dependence structure is unspecified. The focus is on the aggregate risk S = X_1 + … + X_n and its bounds in the sense of convex order. While the sharp upper bound for general n and the sharp lower bound for n=2 are well-known, the sharp lower bound for n ≧ 3 has been a long-standing open problem.

A sharp lower bound for the aggregate risk for n ≧ 3 is derived primarily under the assumption of homogeneous marginal distributions. A specific random variable is constructed and proven to serve as a lower bound for admissible risk in the convex order. Furthermore, it is shown that this lower bound is sharp when the marginal distribution satisfies a condition related to Complete Mixability. The extension of the results to the case of heterogeneous marginal distributions is also discussed.

Bernard, C., Jiang, X., Wang, R. (2014). Risk aggregation with dependence uncertainty. Insurance:Mathematics and Economics, 54, 93–108.