Sharp One-Sided Confidence Bounds Over Positive Regions

The paper develops one-sided analogs to Scheffe's two-sided confidence bounds for a function f (x), x Rⁿ. If the domain X of f is a subset of R_+ⁿ = \x: xᵢ 0, i\, then the upper Scheffe bounds are conservative upper confidence bounds, which can be sharpened, often to great practical advantage. This sharpening, accomplished by a non-trivial extension of Scheffe's method, is developed by the geometry-probability argument of Section 2. Section 3 derives coverage probabilities for general 2- and 3-parameter functions and illustrates savings by the sharp bounds in two examples.