Sharp One-Sided Confidence Bounds Over Positive Regions

Abstract

The paper develops one-sided analogs to Scheffe's two-sided confidence bounds for a function $f(\mathbf{x}), \mathbf{x} \in R^n$. If the domain $X\ast$ of $f$ is a subset of $R_+^n = \{\mathbf{x}: x_i \geqq 0, \forall i\}$, 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.