Probabilistic Scaling for the Numerical Inversion of Nonprobability Transforms

It is known that probability density functions and probability mass functions can usually be calculated quite easily by numerically inverting their transforms (Laplace transforms and generating functions, respectively) with the Fourier-series method, but other more general functions can be substantially more difficult to invert, because the aliasing and roundoff errors tend to be more difficult to control. In this paper we propose a simple new scaling procedure for non-probability functions that is based on transforming the given function into a probability density function or a probability mass function and transforming the point of inversion to the mean. This new scaling is even useful for probability functions, because it enables us to compute very small values at large arguments with controlled relative error. Subject classifications: Mathematics, functions: scaling for numerical transform inversion. Queues, algorithms: scaling for numerical transform inversion.