Examples

This page contains reproducible examples for AdditionalDistributions.jl, a Julia package that extends Distributions.jl with additional probability distributions.

The examples show how to use AdditionalDistributions.jl for simulation, risk modeling, count data, multivariate probabilities, hypothesis testing, copula-based modeling, and Bayesian inference with Turing.jl.

The examples are available in the examples/scripts/ directory of the repository and can be run with:

julia --project=examples examples/scripts/01_distributions_api.jl

To run all examples:

for f in examples/scripts/*.jl; do
    echo ""
    echo "=============================="
    echo "Running $f"
    echo "=============================="
    julia --project=examples "$f"
done

Basic Distributions.jl interface

File:

examples/scripts/01_distributions_api.jl

This example checks that several distributions support the standard Distributions.jl interface, including pdf, logpdf, cdf, quantile, rand, and moments when available.

It includes examples such as:

Lomax(2.5, 1.0)
Burr(2.0, 3.0)
Dagum(2.0, 3.0)
Maxwell(1.5)
Gompertz(1.2, 0.8)
PERT(0.0, 0.6, 1.0)

Heavy-tailed risk models

File:

examples/scripts/02_heavy_tail_risk.jl

This example uses positive heavy-tailed distributions to simulate loss models and estimate empirical risk measures such as Value-at-Risk and Expected Shortfall.

It compares models such as:

Lomax(2.2, 1_000.0)
Burr(1.6, 2.5, 1_000.0)
Dagum(2.0, 1_000.0, 1.5)

Discrete count models

File:

examples/scripts/03_count_models.jl

This example illustrates count distributions such as zero-inflated and heavy-tailed discrete models.

Examples include:

ZIP(2.0, 0.35)
ZINB(5, 0.30, 0.45)
BetaNegBinomial(10, 2.0, 5.0)
Delaporte(2.0, 5.0, 0.45)
Yule(2.5)
Zeta(2.2)

Some of these distributions may have infinite or undefined theoretical moments. In those cases, empirical summaries can be unstable for moderate sample sizes, which is expected for heavy-tailed count models.

Multivariate Gaussian rectangular probabilities

File:

examples/scripts/04_mvgaussian_cdf.jl

This example computes a rectangular probability for a multivariate Gaussian distribution:

res = cdf_result(
    d,
    lower,
    upper;
    rng=rng,
    m=100_000,
    abseps=1e-5,
    releps=1e-5,
)

The returned object includes the estimated probability, an error estimate, the number of evaluations, an algorithm label, and an inform flag.

Multivariate Student-t rectangular probabilities

File:

examples/scripts/05_mvstudent_cdf.jl

This example computes a rectangular probability for a multivariate Student-t distribution using randomized quasi-Monte Carlo integration.

The multivariate Student-t case is handled differently from the Gaussian case because the common radial scale induces dependence even when the scale matrix is diagonal.

HypothesisTests.jl workflows

File:

examples/scripts/06_hypothesistests.jl

This example shows two workflows.

First, it compares two positive heavy-tailed samples using classical univariate tests:

ApproximateTwoSampleKSTest(x, y)
MannWhitneyUTest(x, y)

Second, it uses simulated multivariate Gaussian samples together with multivariate tests from HypothesisTests.jl:

UnequalCovHotellingT2Test(X, Y)
EqualCovHotellingT2Test(X, Y)
BartlettTest(X, Y)

This demonstrates how MvGaussian samples from AdditionalDistributions.jl can be used directly in downstream statistical testing workflows.

Copulas.jl and SklarDist

File:

examples/scripts/07_copulas_sklar.jl

This example builds a joint distribution using a Gaussian copula and marginal distributions from AdditionalDistributions.jl:

C = GaussianCopula(Σ)

marginals = (
    Burr(1.6, 2.5, 1_000.0),
    Dagum(2.0, 1_000.0, 1.5),
)

joint = SklarDist(C, marginals)

It then simulates from the joint model and estimates marginal summaries, rank correlation, and the probability of a joint upper-tail event.

Turing.jl Bayesian model

File:

examples/scripts/08_turing_model.jl

This example uses Maxwell inside a Bayesian model defined with Turing.jl:

@model function maxwell_model(x)
    σ ~ truncated(Normal(1.5, 0.5), 0.1, 5.0)

    for i in eachindex(x)
        x[i] ~ Maxwell(σ)
    end
end

A short NUTS chain recovers the true scale parameter from simulated data, illustrating compatibility with probabilistic programming workflows.