One-sided confidence about functionals over tangent cones

In the setup of i.i.d. observations and a real valued differentiable functional T, locally asymptotic upper bounds are derived for the power of one-sided tests (simple, versus large values of T) and for the confidence probability of lower confidence limits (for the value of T), in the case that the tangent set is only a convex cone. The bounds, and the tests and estimators which achieve the bounds, are based on the projection of the influence curve of the functional on the closed convex cone, as opposed to its closed linear span. The higher efficiency comes along with some weaker, only one-sided, regularity and stability.