AdditionalDistributions.jl

AdditionalDistributions.jl extends the Distributions.jl ecosystem with additional continuous, discrete, and multivariate probability distributions.

The package follows the standard Distributions.jl interface whenever possible and also provides native Julia routines for rectangular probabilities of multivariate Gaussian and Student-t distributions.

Installation

AdditionalDistributions.jl is registered in the Julia General registry:

using Pkg
Pkg.add("AdditionalDistributions")

Then:

using AdditionalDistributions

Highlights

• Additional continuous and discrete probability distributions.
• Zero-inflated, heavy-tailed, reliability, and count models.
• MvGaussian integrated with the Distributions.AbstractMvNormal interface.
• MvTStudent built around native Distributions.jl Student-t distributions.
• Rectangular Gaussian and Student-t CDF evaluation through randomized QMC.
• Direct cdf_result support for native MvNormal and MvTDist objects.
• Structured numerical diagnostics through CDFResult.
• Reproducible randomized integration with seeded RNGs.
• Integration examples with the wider Julia statistics ecosystem.

Quick start

using AdditionalDistributions
using Distributions

d = Lomax(2.0, 3.0)

pdf(d, 1.5)
cdf(d, 1.5)
quantile(d, 0.9)
rand(d)

Discrete distributions use the same interface:

d = ZIP(2.0, 0.3)

pdf(d, 0)
cdf(d, 4)
rand(d)

See the distribution index for the complete public API.

Multivariate rectangular probabilities

For a multivariate random vector (X), the package evaluates probabilities of the form

$$P(a_i \le X_i \le b_i,\quad i=1,\ldots,d).$$

using AdditionalDistributions
using LinearAlgebra
using Random

d = 10
Σ = fill(0.5, d, d)
Σ[diagind(Σ)] .= 1.0

lower = fill(-1.0, d)
upper = fill(1.0, d)

dist = MvGaussian(zeros(d), Σ)
res = cdf_result(dist, lower, upper; m=100_000, rng=MersenneTwister(1234))

res.value      # probability estimate
res.error      # estimated integration error
res.inform     # integration status
res.neval      # requested integration budget
res.algorithm  # algorithm identifier

For package-owned wrappers, cdf(dist, lower, upper) returns only the probability estimate.

The same numerical backend can also be used directly with native Distributions.jl objects:

using Distributions

dn = MvNormal(zeros(d), Σ)
dt = MvTDist(4.0, zeros(d), Σ)

cdf_result(dn, lower, upper; rng=MersenneTwister(1234))
cdf_result(dt, lower, upper; rng=MersenneTwister(1234))

The package deliberately exposes this interoperability through cdf_result rather than adding Distributions.cdf methods to external distribution types.

Numerical methods

Multivariate rectangular probabilities use Genz-style variable conditioning combined with randomized rank-1 lattice quasi-Monte Carlo integration.

The floating-point path uses cached component-by-component lattice construction and tent transformation. Gaussian and Student-t implementations use specialized integration dimensions and Student-t radial transforms where applicable.

Accuracy is assessed against deterministic structured reference probabilities when available. Independent implementations such as MvNormalCDF.jl, SciPy, and R's mvtnorm are used for comparison rather than as ground truth.

See:

• Accuracy and reproducibility
• Benchmarks
• Multivariate distributions

Running tests

julia --project=. -e 'using Pkg; Pkg.test()'

Contributing

Contributions are welcome. See CONTRIBUTING.md for development conventions and guidelines.

Citation

If you use AdditionalDistributions.jl in your research, please cite:

S. Jiménez (2025). AdditionalDistributions.jl — Advanced and Extended Probability Distributions in Julia.

@misc{Jimenez2025AdditionalDistributions,
    author = {Santiago Jimenez},
    title = {AdditionalDistributions.jl --- Advanced and Extended Probability Distributions in Julia},
    year = {2025},
    url = {https://github.com/Santymax98/AdditionalDistributions.jl},
    note = {Julia package}
}

License

MIT License.