Volumes of Tubular Neighborhoods of Spherical Polyhedra and Statistical Inference

Abstract

For statistical procedures including Scheffe-type simultaneous confidence bounds for response surfaces and likelihood ratio tests for an additional regressor with unspecified parameters in a regression model, the confidence level or size can be expressed in terms of probabilities of the form $P\lbrack U \in D(\Gamma, \theta)\rbrack$, where $\Gamma$ is a subset of $S^m$ (the unit sphere in $R^{m + 1}), U$ is uniformly distributed in $S^m$ and $D(\Gamma, \theta)$ denotes the tubular neighborhood of $\Gamma$ of angular radius $\theta$, the set of points in $S^m$ whose angular distance from $\Gamma$ is at most $\theta$. Consequently, determining critical points involves the calculation of the volumes of tubes. For the case when $\Gamma$ is the diffeomorphic image of an $r$-dimensional convex polytope, an upper bound is given for the volume of its tubular neighborhood when the tube radius is sufficiently small, and which is exact in some special cases. Even if the tubular radius is moderate in size, the expression can be used to approximate the volume. The volume expression is a sum of $r$-fold integrals, one corresponding to each face of the polytope, and is derived using a result of Weyl (1939), which gives the volume of a tubular neighborhood of a $k$-dimensional submanifold of the unit sphere. Use of the expression leads to conservative statistical procedures when the desired error probability is sufficiently small and to asymptotically valid procedures as the error probability goes to zero.