Precise quantile function estimation from the characteristic function
Gero Junike
TL;DR
This work tackles precise estimation of the quantile function $F^{-1}$ from a known characteristic function $\varphi$ for continuous distributions. It introduces the COS method to construct an error-controlled CDF approximation $H_{COS}$ from $\varphi$, and derives how the CDF error propagates to the QF via an inverse-function bound, with explicit parameter choices for $(a,b,N)$ and robust inversion. Theoretical results show that the QF error depends linearly on the CDF approximation error and inversely on the derivative $h=H'$ in the tails, with exponential convergence guaranteed under semi-heavy tails, and are backed by numerical experiments on NIG and TS distributions demonstrating accuracy, speed, and practical error control. The methodology enables high-precision, QF-based random number generation for distributions lacking closed-form densities or efficient samplers, with explicit error budgets and fast computation.
Abstract
We provide theoretical error bounds for the accurate numerical computation of the quantile function given the characteristic function of a continuous random variable. We show theoretically and empirically that the numerical error of the quantile function is typically several orders of magnitude larger than the numerical error of the cumulative distribution function for probabilities close to zero or one. We introduce the COS method for computing the quantile function. This method converges exponentially when the density is smooth and has semi-heavy tails and all parameters necessary to tune the COS method are given explicitly. Finally, we numerically test our theoretical results on the normal-inverse Gaussian and the tempered stable distributions.