Precise quantile function estimation from the characteristic function

TL;DR

This paper introduces a method for accurately estimating the quantile function from the characteristic function, providing theoretical error bounds and demonstrating the method's effectiveness on complex distributions.

Contribution

It presents the COS method for quantile function computation, with explicit parameter tuning and proven exponential convergence for smooth, semi-heavy tail distributions.

Findings

Numerical error in quantile estimation can be significantly larger than in CDF estimation near distribution tails.

The COS method converges exponentially for smooth densities with semi-heavy tails.

Empirical tests on normal-inverse Gaussian and tempered stable distributions validate the theoretical bounds.

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.