A shape-preserving spline interpolation for sampling designs from inverse distributions

Abstract

Given a strictly monotone cumulative function F:[a,b]⟶[c,d], with a,b,c,d∈R, a<band c<d, such that F[a,b]=[c,d]. The possibility of using the spline interpolation procedure, to approximate the inverse of F, is a natural way in some applications, and it leads to schemes that are stable and feasible to implement. We prove that, with a shape-preserving quadratic Hermite interpolation scheme based on quadratic B-splines, we can preserve the monotonicity of the inverse of F while maintaining a third-order estimate of the interpolation error. To illustrate the effectiveness of this approach, we provide some examples of application, such as the inversion of cumulative distribution functions (arc-length and normal cumulative distributions) and the computation of the Lambert W-function.

Keywords

• Inversion of functions
• Monotone and convex interpolation
• Inverse of the arc-length function
• Inverse of the normal cumulative distribution
• Lambert W-function