Add Pareto Type II (Lomax) distribution

[Copilot]

Implements the Pareto Type II distribution corresponding to GAMLSS PARETO2() family.

Implementation

• Distribution class: Pareto using ScipyMixin with scipy.stats.lomax
• Parameterization: GAMLSS (μ, σ) → scipy (scale=μ, c=σ, loc=0)
• Derivatives: Observed information (depends on y)
  • First: dl/dμ = (σy - μ)/(μ(μ+y)), dl/dσ = 1/σ - log(1 + y/μ)
  • Second: d²l/dμ² = (μ² - 2μσy - σy²)/(μ²(μ+y)²), d²l/dσ² = -1/σ²
  • Cross: d²l/dμdσ = y/(μ(μ+y))

Files

• src/ondil/distributions/pareto.py - Main implementation
• src/ondil/distributions/__init__.py - Export added

Usage

from ondil.distributions import Pareto
from ondil.estimators import OnlineDistributionalRegression

estimator = OnlineDistributionalRegression(
    distribution=Pareto(),
    equation={0: np.array([0, 1]), 1: np.array([0, 1])},
    method="ols"
)
estimator.fit(X=X, y=y)

PDF: f(y|μ,σ) = σ/(μ(1 + y/μ)^(σ+1)) for y ≥ 0, μ > 0, σ > 0

• Fixes Add distribution: Pareto #108

Original prompt

---

This section details on the original issue you should resolve

<issue_title>Add distribution: Pareto</issue_title>
<issue_description>https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lomax.html#scipy.stats.lomax</issue_description>

<agent_instructions>Please develop the distribution.

• Use the scipyMixin class to re-use scipy infrastructure.
• Use the generalized tests in tests/distribution/test_xxx.py to test the CDF, PPF, .... and the derivatives.
• Use the skew-t and the normal distributions as example. Use case match in favour of older if -else code. Use the gama distribution as example for mapping scipy and gamlss parameterizations if they differ.

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lomax.html#scipy.stats.pareto

https://github.com/gamlss-dev/gamlss.dist/blob/main/R/PARETO2.R</agent_instructions>

Comments on the Issue (you are @copilot in this section)

• Fixes Add distribution: Pareto #108

---

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[simon-hirsch]

@copilot the derivatives seem not to match. Please look at the output of the automated tests.

@copilot please dont implement a test for the coefficients manually.

[Copilot]

@copilot the derivatives seem not to match. Please look at the output of the automated tests.

@copilot please dont implement a test for the coefficients manually.

Fixed the derivatives to use observed information instead of expected Fisher information, matching GAMLSS PARETO2(). Also removed the manual coefficient test files as requested. Changes in commit b075bbf.