On the relationship between bounds on the tails of compound distributions

A comparison of two upper bounds for the tails of compound distributions, both defined in terms of a new worse than used (NWU) distribution, reveals that one is sharper in the decreasing failure rate (DFR) case. An inductive argument is employed to construct a lower bound in terms of a new better than used (NBU) distribution which is a dual to the upper bound. It is also sharper than the corresponding lower bound in the increasing failure rate (IFR) case. Applications to ruin theory are then given.