The volume-of-tubes formula, originally developed by Hotelling and
Weyl in 1939, computes the volumes of tubular neigbourhoods of
manifolds. Given a suitable set (curve, surface e.t.c.) lying in
n-dimensional space, the tube formula gives an expression for
the volume of the set of all points within a specified radius of the set.
With some additional mathematics, the tube formula provides accurate
distributional approximations for the extreme values of Gaussian and other
stochastic processes.
The tube formula can be applied to many statistical problems where
such extreme values arise. Applications are particularly numerous
in the area of simultaneous inference and testing.
Despite both the mathematical beauty and simplicity of the end results,
the method remains underutilized in statistics, with
more computational, less accurate and non-reproducible simulations
often being the prefered choice. The libtube library
is a linkable library implementing the volumes-of-tube formula, distributed
with the aim of making the formula more accessible to statistical researchers
in their work.
Statistical problems to which the tube formula can be applied include
- Significance testing in nonlinear regression.
- Simultaneous confidence bands for linear models and smoothing.
- Functional linear models and analysis-of-variance.
- Testing problems involving scan statistics and mixture models.
- Order-restricted inference.