Welcome to AdditionalDistributions.jl ​

The AdditionalDistributions.jl package is a comprehensive extension of Distributions.jl, designed to expand its functionality by incorporating both discrete, continuous, and multivariate probability distributions not available in the base package. It provides a unified, research-grade framework for probabilistic modeling, simulation, and numerical evaluation of advanced statistical distributions.

Overview ​

With AdditionalDistributions, you can:

• Sample from distributions: Draw random samples from classical and novel models.

• Compute densities and probabilities: Access pdf, cdf, and logpdf for a wide range of families.

• Evaluate multivariate probabilities: Compute rectangular probabilities for Gaussian and Student’s t models using a high-performance QMC integrator.

• Calculate statistical properties: Obtain mean, variance, skewness, kurtosis, entropy, and characteristic functions.

• Extend or define new distributions: Create custom statistical models compatible with Distributions.jl’s interface.

Highlights ​

Motivation ​

Many probability distributions and accurate multivariate CDFs remain unavailable or fragmented across Julia packages. AdditionalDistributions.jl aims to unify them under a consistent API, enabling both theoretical and applied work in statistics, econometrics, and data science.

The package is also a testbed for algorithmic optimization — including Quasi-Monte-Carlo integration, low-discrepancy sequences, and vectorized numerical transforms — with the goal of eventually contributing mature implementations back to Distributions.jl.

Results (illustrative comparison) ​

Representative results under identical conditions (same machine, same Σ, same limits [a,b], same m, fixed seed):

For Student’s t (e.g., d=5, ν=10, m=20 000), typical outputs are of the form:

p ≈ 0.2468,  error ≈ 8.2e-5,  inform = 0

Interpretation

• QMC yields reproducible results (fixed seed) and competitive accuracy.

• Reported errors are estimators, not strict bounds; they vary with m, dimension, correlation, and region [a,b].

• In some problems Genz–Bretz reports smaller errors for the same m; in others QMC matches or surpasses. The goal here is transparent, honest comparability.

Usage snippets ​

Multivariate Gaussian (rectangular CDF) ​

using AdditionalDistributions

μ = zeros(5)
Σ = [1.0 0.6 0.3 0.1 0.0;
     0.6 1.0 0.5 0.3 0.1;
     0.3 0.5 1.0 0.6 0.3;
     0.1 0.3 0.6 1.0 0.6;
     0.0 0.1 0.3 0.6 1.0]

a = fill(-1.0, 5)
b = fill( 1.0, 5)

d = MvGaussian(μ, Σ)
p = cdf(d, a, b; m=50_000)                 # or full=true to get (value, error, inform)

Multivariate Student’s t (rectangular CDF) ​

ν = 10
Σ = [1.0 0.4 0.2 0.1 0.05;
     0.4 1.0 0.5 0.2 0.1;
     0.2 0.5 1.0 0.4 0.2;
     0.1 0.2 0.4 1.0 0.3;
     0.05 0.1 0.2 0.3 1.0]

a = fill(-0.5, 5)
b = fill( 0.5, 5)

d_t = MvTStudent(ν, zeros(5), Σ)
val, err, info = cdf(d_t, a, b; m=20_000, full=true)

Methodological notes ​

• Algorithms:

  • MvGaussian / MvTStudent use QMC with Richtmyer lattices + Cranley–Patterson shifts; optional antithetic reflection in Gaussian coordinates; reproducibility via seeded RNG.

• Parameters that matter:

  • Budget m, dimension d, correlation structure, geometry of [a,b]. For Student’s t, also ν and the standardization to correlation.

• Reporting:

  • full=true returns (value, error, inform). inform = 1 indicates tolerance not met under the current m; increase m or relax tolerances.

Roadmap ​

• Extend coverage to Generalized Hyperbolic, Skew-t, Variance-Gamma.

• Add parameter fitting (fit_mle, fit_map) for new families.

• Explore GPU-assisted QMC for high-dimensional CDFs.

• Improve diagnostics (inform) and optional convergence logging.

• Publish reproducible comparisons across Julia and R ecosystems.

Contributing ​

We follow the development guidelines and testing conventions of Distributions.jl:

• Use @testitem blocks with explicit tolerances and seeded randomness when applicable.

• Include doctest examples in docstrings.

• Keep methods type-stable, memory-aware, and numerically robust.

Ways to contribute

1. Report issues with clear minimal reproductions.

2. Propose improvements (performance, modularity, memory) via PRs.

3. Discuss algorithms (QMC variance reduction, pivots, transforms).

4. Documentation (examples, edge cases, methodological notes).

Related ecosystem ​

• Distributions.jl — Base statistical library.

• MvNormalCDF.jl — Genz–Bretz MVN CDF.

• StatsBase.jl — Foundational statistics.

Citation ​

If you use AdditionalDistributions.jl in academic work, please cite:

S. Jiménez (2025). AdditionalDistributions.jl — Advanced and Extended Probability Distributions in Julia.
Available at: https://github.com/Santymax98/AdditionalDistributions.jl

Author: Santiago Jiménez License: MIT Repository: Santymax98/AdditionalDistributions.jl