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On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution

Guillermo Navas-Palencia

TL;DR

This work develops a comprehensive framework for computing the Normal Inverse Gaussian CDF by deriving diverse series and asymptotic expansions from Laplace-type representations, supplemented with reliable numerical integration. It covers special cases ($\beta=0$ and $x=\mu$) and the general case, including uniform and accelerated expansions that maintain accuracy across broad parameter regimes. The authors implement these methods in C++, provide rigorous error considerations, and demonstrate substantial speedups (often 5–60×) over existing adaptive numerical integration in open-source libraries while maintaining near-machine precision. The practical impact is a robust, fast, and versatile tool for NIG CDF computation, with potential extensions to related GH-family distributions and broader numerical applications.

Abstract

In this paper, we obtain various series and asymptotic expansions involving the modified Bessel function of the second kind for the normal inverse Gaussian cumulative distribution function. The new expansions accelerate computations, complementing the numerical integration methods implemented in statistical software packages. We also provide a detailed description of the algorithm and its corresponding implementation in C++. The performance and accuracy of the algorithm are extensively tested and benchmarked with open-source implementations, offering superior accuracy and speed-ups of a factor from 5 to 60.

On the computation of the cumulative distribution function of the Normal Inverse Gaussian distribution

TL;DR

This work develops a comprehensive framework for computing the Normal Inverse Gaussian CDF by deriving diverse series and asymptotic expansions from Laplace-type representations, supplemented with reliable numerical integration. It covers special cases ( and ) and the general case, including uniform and accelerated expansions that maintain accuracy across broad parameter regimes. The authors implement these methods in C++, provide rigorous error considerations, and demonstrate substantial speedups (often 5–60×) over existing adaptive numerical integration in open-source libraries while maintaining near-machine precision. The practical impact is a robust, fast, and versatile tool for NIG CDF computation, with potential extensions to related GH-family distributions and broader numerical applications.

Abstract

In this paper, we obtain various series and asymptotic expansions involving the modified Bessel function of the second kind for the normal inverse Gaussian cumulative distribution function. The new expansions accelerate computations, complementing the numerical integration methods implemented in statistical software packages. We also provide a detailed description of the algorithm and its corresponding implementation in C++. The performance and accuracy of the algorithm are extensively tested and benchmarked with open-source implementations, offering superior accuracy and speed-ups of a factor from 5 to 60.

Paper Structure

This paper contains 47 sections, 5 theorems, 191 equations, 20 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Distribution properties
  3. Density function
  4. Cumulative distribution function
  5. Alternative integral representations
  6. Methods of computation
  7. Special case $\beta = 0$
  8. Expansions $|x-\mu| \to 0$
  9. Convergence acceleration of the expansion $|x-\mu| \to 0$
  10. Expansion $|x-\mu| \to \infty$
  11. Uniform expansion $\alpha \to \infty$, $\alpha \sim \delta$ and $|x-\mu| \gg 0$
  12. Special case $x = \mu$
  13. General case
  14. Expansions $|x-\mu| \to 0$ in terms of modified Bessel functions
  15. Expansion $|x-\mu| \to 0$ in terms of modified Bessel and incomplete gamma functions
  16. ...and 32 more sections

Key Result

Proposition 2.1

An incomplete Laplace-type integral representation in terms of modified Bessel function $K_0(x)$ is given by

Theorems & Definitions (7)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4