The Falling Factorial Basis and Its Statistical Applications
This work addresses computational bottlenecks in statistical modeling for researchers and practitioners, though it is incremental as it builds on existing spline methods.
The paper introduces the falling factorial basis, a spline-like basis enabling linear-time computations for matrix operations, and applies it to trend filtering and a higher-order two-sample Kolmogorov-Smirnov test, showing efficiency gains.
We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factorial functions are not actually splines, but are close enough to splines that they provably retain some of the favorable properties of the latter functions. We examine their application in two problems: trend filtering over arbitrary input points, and a higher-order variant of the two-sample Kolmogorov-Smirnov test.