Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics

Abstract

Stable distributions are an important class of infinitely divisible probability distributions, of which two special cases are the Cauchy distribution and the normal distribution. Aside from a few special cases, the density function for stable distributions has no known analytic form and is expressible only through the variate’s characteristic function or other integral forms. In this paper, we present numerical schemes for evaluating the density function for stable distributions, its gradient, and distribution function in various parameter regimes of interest, some of which had no preexisting efficient method for their computation. The novel evaluation schemes consist of optimized generalized Gaussian quadrature rules for integral representations of the density function, complemented by asymptotic expansions near various values of the shape and argument parameters. We report several numerical examples illustrating the efficiency of our methods. The resulting code has been made available online.

Appendix: Gradients of series expansions

Here we provide formulae for the derivatives of the series expansions presented in Sect. 2.4. From (2.18),

$$\begin{aligned}&\partial _x f(x;\alpha , \beta ) \nonumber \\&\quad = \frac{1}{\alpha \pi } \sum _{k=0}^\infty \frac{ \Gamma ( \frac{k+2}{\alpha } )}{ \Gamma ( k )} (1+\zeta ^2)^{ - \frac{k+2}{ 2\alpha }}\nonumber \\&\qquad \times \sin \left( \left( \pi /2 + (\arctan \zeta )/\alpha \right) (k+2) \right) (x - \zeta )^{k}. \end{aligned}$$

(5.1)

Using an error bound analogous to the ones in Sect. 2.3, we have that

$$\begin{aligned} |x-\zeta | \le C_n^0(\alpha ) := \left[ \epsilon \alpha \pi (1+\zeta ^2)^{\frac{n+2}{2\alpha }} \frac{\Gamma (n)}{\Gamma ( \frac{n+2}{\alpha })} \right] ^{1/n}. \end{aligned}$$

(5.2)

By differentiating (2.25), we attain

$$\begin{aligned} \partial _x f(x,\alpha , \beta )= & {} \frac{\alpha }{\pi } \sum _{k=1}^\infty (-1)^{k} \frac{ (\alpha k + 1) \Gamma (\alpha k)}{\Gamma (k)} (1+\zeta ^2)^{k/2} \nonumber \\&\times \sin ((\pi \alpha - \arctan \zeta ) k) (x-\zeta )^{-\alpha k-2},\nonumber \\ \end{aligned}$$

(5.3)

whose radius of convergence to precision $\epsilon$ we estimate by

$$\begin{aligned}&|x-\zeta | \ge C_{n-1}^\infty (\alpha )\nonumber \\&\quad := \left[ \frac{\alpha }{\pi \epsilon } (1+\zeta ^2)^{\frac{n}{2} } \frac{ (\alpha n + 1)\Gamma (\alpha n)}{\Gamma ( n ) } \right] ^{1/(\alpha n-2)}. \end{aligned}$$

(5.4)

For the parameter ranges where there is no convenient formulation of the derivatives, we can use a finite difference approximation of the form

$$\begin{aligned}&\partial _x f(x)\nonumber \\= & {} \frac{ -f(x+2h) + 8 f(x+h) - 8 f(x-h) + f(x-2h) }{12h} \nonumber \\&+ \, O(h^4). \end{aligned}$$

(5.5)