StableDistributions.jl

Generation and estimation of the class of stable distributions (see on Wikipedia) in Julia. It fully complies with the interface of Distributions.jl package. The algorithms are based on book John P. Nolan, "Univariate Stable Distributions", Springer 2020 and related publications of John P. Nolan.

Defining

This package uses the so-called type-1 parametrisation determined by:

• stability index 0 < α ≤ 2,
• skewness parameter -1 ≤ β ≤ 1,
• scale 0 < σ,
• location μ.

Such distribution is uniquely characterised by its characteristic function (Fourier transform of its pdf)

$$\varphi(t; \alpha, \beta, \sigma, \mu) = \exp\big(\mathrm{i}t\mu -|\sigma t|^\alpha(1-\mathrm i\beta\mathrm{sgn}(t)\Phi(t))\big)$$

with $\Phi(t) = -\frac{2}{\pi}\log|t|$ for α = 1 or $\Phi(t) = \tan(\pi\alpha/2)$ for α ≠ 1.

To construct stable distribution one can use:

• Stable(α) for standard symmetric α-stable distribution equivalent to Stable(α, 0, 1, 0),
• Stable(α, β) for standard α-stable distribution with skewness parameter β equivalent to Stable(α, β, 1, 0),
• Stable(α, β, σ, μ) in general case.

Sampling

One can use standard functions rand(d::Stable) for one value or rand(d::Stable, shape) for a series of values formatted with a given shape.

Features

Probability density function pdf, comultative probability function cdf, moment generating function mgf, characteristic function cf, quantiles quantile, and few ralated functions are also available. Values of pdf, cdf and quantile are approximate and based on numerical approximations of the corresponding integral representions and additional numerical function inversion for quantile. Maximum of the pdf can be numerically approximated using mode. For example, pdf(Stable(1.5), 2) returns pdf of Stable(1.5) at point x = 2 which is approximately 0.084.

Transforming

One can multiply stable distributions by scalars and add them, e.g. 2Stable(1.5) + 3 is a valid code which returns Stable(1.5, 0, 2, 3). Function convolve is used to convolve two stable distributions with the same α, which is a distribution of the sum of two indepedent stable variables with such distributions. For example convolve(Stable(0.5,1), Stable(0.5,0,2,1)) gives approximately Stable(0.5, 0.41, 5.83, 1). Function support returns support of a given stable distribution, which can be half-bounded for skewed stable distribution with β = 1 or β = -1 and α < 1.

Method convert can be used to convert from special cases of Lévy, Cauchy and Gaussian to stable distributions. For example, convert(Stable,Normal(0,1)) returns equivalent of Stable(2, 0, √2/2, 1).

Fitting

Given data sample one can find best fitting stable distribution using method fit(Stable, sample). It uses algorithm based on finding parameters which best fit sample characteristic function. This methods is considered to be efficient and quick. Alternatively, one can use fit_quantile(Stable, sample) which uses older McCulloch's quantile method in type-0 parametrisation. This method is considered to be generally worse but can be useful for additional checks, e.g. when there are doubts about the reliability of the estimation.