lossmodels

A Python library for actuarial loss modeling using frequency–severity methods.

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Overview

lossmodels is a readable, NumPy/SciPy-backed implementation of the core loss-modeling toolkit: a large catalog of severity and frequency distributions, aggregate (collective-risk) loss models, coverage modifications, and fitting and model-selection utilities. It is aimed at actuarial students, analysts, insurance data scientists, and quantitative developers who want a lightweight library whose parameterizations line up with the standard references.

The continuous-severity and discrete-frequency inventories use the standard scale-parameterized forms common in actuarial loss modeling, so means, moments, and quantiles line up with the formulas you would compute by hand.

Highlights

• Severity models — a comprehensive continuous inventory: lognormal, gamma, Weibull, exponential and their inverses; the transformed-beta family (Burr, inverse Burr, generalized Pareto, Pareto/Lomax, inverse Pareto, loglogistic, paralogistic, inverse paralogistic); inverse Gaussian, log-t, single-parameter Pareto; and finite-support beta / generalized beta.
• Frequency models — the (a, b, 0) class (Poisson, binomial, geometric, negative binomial) and the (a, b, 1) class (zero-truncated, zero-modified, logarithmic).
• Spliced / composite severities — a body distribution below a threshold and a heavier tail above it, behind a single severity interface.
• Empirical distributions for raw frequency and severity data.
• Aggregate loss modeling — Monte Carlo simulation, Panjer recursion, and FFT aggregation, plus distribution-level and PMF-level risk measures (VaR, TVaR, stop-loss, limited expected value).
• Coverage modifications — ordinary deductibles, policy limits, and layers, each wrapping any severity and re-usable everywhere a severity is accepted.
• Estimation — maximum likelihood and method-of-moments fitters, log-likelihood / AIC / BIC diagnostics, goodness-of-fit statistics (KS, Anderson–Darling, Cramér–von Mises), and automatic best-fit selection.

Installation

From PyPI:

pip install lossmodels

In development mode from a source checkout:

pip install -e .

Requires Python >=3.10; the only runtime dependencies are numpy and scipy.

Package structure

lossmodels/
├── severity/      # continuous claim-severity distributions
├── frequency/     # discrete claim-count distributions
├── aggregate/     # collective-risk models, discretization, Panjer, FFT, risk metrics
├── coverage/      # deductibles, limits, and layers
├── empirical/     # empirical frequency and severity distributions
├── estimation/    # MLE, method of moments, diagnostics, model selection
└── utils/         # shared numeric / validation helpers

Quick start

A frequency–severity aggregate model

from lossmodels import Poisson, Lognormal, CollectiveRiskModel

freq = Poisson(lam=2.0)
sev = Lognormal(mu=10.0, sigma=0.8)

model = CollectiveRiskModel(freq, sev)

print("Mean:", model.mean())
print("Variance:", model.variance())
print("VaR 95%:", model.var(0.95))
print("TVaR 95%:", model.tvar(0.95))

samples = model.sample(50_000)
print("Simulated mean:", samples.mean())

Every model name is importable directly from the top level (from lossmodels import Burr, ZeroTruncated, ...) or from its submodule (from lossmodels.severity import Burr).

Using the extended distribution catalog

from lossmodels import Burr, ParetoII, ZeroTruncated, Poisson, CollectiveRiskModel

sev = ParetoII(alpha=3.0, theta=1000.0)        # the two-parameter "Pareto" (Type II / Lomax)
freq = ZeroTruncated(Poisson(2.0))             # an (a, b, 1) frequency

model = CollectiveRiskModel(freq, sev)
print(model.mean(), model.var(0.99))

burr = Burr(alpha=3.0, theta=1000.0, gamma=2.0)
print(burr.mean(), burr.cdf(2500.0), burr.quantile(0.95))
print(burr.limited_expected_value(1000.0))     # E[min(X, 1000)]

Severity models

The shared severity interface

Every severity exposes the same interface, so they are interchangeable across the aggregate, coverage, and estimation tooling:

Method	Meaning
pdf(x), cdf(x)	density and distribution function (scalar or array)
quantile(p) / ppf(p)	inverse CDF / Value-at-Risk
sample(size)	random variates
mean(), variance(), std()	moments (raise where they do not exist)
limited_expected_value(d)	E[min(X, d)]
excess_loss(d)	E[(X − d)₊], the expected cost per loss above d

Moments are returned from closed-form expressions and raise a ValueError outside their existence range (e.g. a Pareto mean with alpha ≤ 1), rather than silently returning a wrong or infinite number. A few distributions (inverse Pareto, inverse exponential, log-t) have no finite positive moments, so their mean/variance always raise.

Base and classic distributions

Distribution	Constructor	Support
Exponential	Exponential(rate)	x > 0
Gamma	Gamma(alpha, theta)	x > 0
Lognormal	Lognormal(mu, sigma)	x > 0
Weibull	Weibull(k, lam)	x > 0
Pareto (Type I)	Pareto(alpha, theta)	x ≥ theta

Transformed beta family

Distribution	Constructor
Burr (Type XII / Singh–Maddala)	Burr(alpha, theta, gamma)
Inverse Burr (Dagum)	InverseBurr(tau, theta, gamma)
Generalized Pareto (transformed-beta)	GeneralizedPareto(alpha, theta, tau)
Pareto (Type II / Lomax)	ParetoII(alpha, theta)
Inverse Pareto	InversePareto(tau, theta)
Loglogistic (Fisk)	Loglogistic(gamma, theta)
Paralogistic	Paralogistic(alpha, theta)
Inverse paralogistic	InverseParalogistic(tau, theta)

Transformed gamma family

Distribution	Constructor
Inverse gamma (Vinci)	InverseGamma(alpha, theta)
Inverse Weibull (log-Gompertz)	InverseWeibull(theta, tau)
Inverse exponential	InverseExponential(theta)

Other distributions

Distribution	Constructor
Inverse Gaussian (Wald)	InverseGaussian(mu, theta)
Log-t	LogT(r, mu, sigma)
Single-parameter Pareto (Type I)	SingleParameterPareto(alpha, theta)

Finite-support distributions

Distribution	Constructor	Support
Beta	Beta(a, b, theta)	0 < x < theta
Generalized beta	GeneralizedBeta(a, b, theta, tau)	0 < x < theta

Parameterization notes

A few naming points matter:

• Two Paretos. ParetoII(alpha, theta) is the two-parameter "Pareto" (Type II / Lomax, support x > 0). The classic Pareto(alpha, theta) is the Type I distribution (support x ≥ theta), commonly called the single-parameter Pareto; it is also exposed under that name as SingleParameterPareto.
• GeneralizedPareto here is the three-parameter transformed-beta distribution, not the extreme-value GPD used in peaks-over-threshold tail fitting. The extreme-value distributions (GEV, Gumbel, Fréchet, GPD, Hill) live in the companion extremeloss package.
• Exponential is parameterized by rate (= 1/theta) and Weibull by shape k and scale lam; the transformed-beta and transformed-gamma families follow the scale-parameter (theta) convention.

Frequency models

The shared frequency interface

Every frequency model exposes pmf(k), cdf(k), sample(size), mean(), variance(), and std().

The (a, b, 0) class

Distribution	Constructor
Poisson	Poisson(lam)
Binomial	Binomial(n, p)
Geometric	Geometric(p)
Negative binomial	NegativeBinomial(r, p)

The (a, b, 1) class

The (a, b, 1) distributions are built from a base (a, b, 0) model:

from lossmodels import Poisson, NegativeBinomial, ZeroTruncated, ZeroModified, Logarithmic

zt = ZeroTruncated(Poisson(2.0))                 # zero is impossible
zm = ZeroModified(NegativeBinomial(3.0, 1/3), p0_modified=0.4)   # custom mass at zero
lg = Logarithmic(beta=3.0)                       # no (a, b, 0) parent

print(zt.mean(), zt.pmf(0))   # 0.0 mass at zero
print(zm.mean(), zm.pmf(0))   # 0.4 at zero
print(lg.mean())

Distribution	Constructor	Notes
Zero-truncated	ZeroTruncated(base)	removes the mass at zero of any (a, b, 0) model and renormalizes
Zero-modified	ZeroModified(base, p0_modified)	places an arbitrary probability p0_modified ∈ [0, 1) at zero
Logarithmic	Logarithmic(beta)	the limiting member with no (a, b, 0) parent

The wrappers respect the base model's probability parameterization; only Logarithmic uses beta. For a beta-parameterized geometric or negative binomial, set p = 1 / (1 + beta).

Spliced severities

SplicedSeverity joins a body distribution (renormalized below the threshold) to a tail distribution supported above it. The tail must satisfy cdf(threshold) = 0 (e.g. a Type I Pareto with theta equal to the threshold), and the mixing weight is the probability mass assigned to the body.

from lossmodels import Lognormal, Pareto, SplicedSeverity

u = 50_000
body = Lognormal(mu=10.0, sigma=0.8)
tail = Pareto(alpha=2.5, theta=u)     # Type I tail, cdf(u) = 0

spliced = SplicedSeverity(body=body, tail=tail, threshold=u, weight=body.cdf(u))
print(spliced.mean(), spliced.quantile(0.99))

The result is itself a severity (pdf, cdf, quantile, sample, mean, variance), so a tail-corrected severity drops straight back into the aggregate and coverage tooling.

Empirical distributions

from lossmodels import EmpiricalSeverity, EmpiricalFrequency

sev = EmpiricalSeverity(claim_amounts)
freq = EmpiricalFrequency(claim_counts)

Both expose the standard severity / frequency interface built from the raw data.

Coverage modifications

Deductibles, limits, and layers each wrap a severity and are themselves severities, so they compose and feed any aggregate model.

from lossmodels import Lognormal, Poisson, CollectiveRiskModel
from lossmodels.coverage import OrdinaryDeductible, PolicyLimit, Layer

base_sev = Lognormal(mu=10.0, sigma=0.8)

with_deductible = OrdinaryDeductible(base_sev, d=10_000)
with_limit = PolicyLimit(base_sev, u=50_000)
layer = Layer(base_sev, d=10_000, u=40_000)     # pays the loss in the layer above d up to u

model = CollectiveRiskModel(Poisson(lam=2.0), layer)
print(model.mean())

Modification	Constructor
Ordinary deductible	OrdinaryDeductible(severity, d)
Policy limit	PolicyLimit(severity, u)
Layer	Layer(severity, d, u)

Aggregate loss modeling

The collective-risk model

CollectiveRiskModel(frequency, severity) is the main entry point. It exposes:

Method	Meaning
mean(), variance(), std()	aggregate moments
var(q), tvar(q)	Value-at-Risk and Tail-VaR at level q
stop_loss(d)	stop-loss premium E[(S − d)₊]
limited_expected_value(d)	E[min(S, d)]
frequency_mean(), severity_mean()	component means
sample(size)	simulate aggregate losses
summary()	a one-call overview of the model

Panjer recursion and FFT

For exact (discretized) aggregate distributions, discretize the severity and then recurse or transform:

from lossmodels import Poisson, Lognormal
from lossmodels.aggregate import (
    discretize_severity, panjer_recursion, fft_aggregate_poisson,
)

freq = Poisson(lam=3.0)
sev = Lognormal(mu=8.0, sigma=0.7)

sev_pmf = discretize_severity(sev, h=100.0, max_loss=500_000.0)
agg_pmf = panjer_recursion(freq, sev_pmf, n_steps=5000)
# FFT aggregation (Poisson frequency):
agg_pmf_fft = fft_aggregate_poisson(freq, sev_pmf, n_steps=5000)

Risk measures

Distribution-level helpers operate on a simulated loss vector, and PMF-level helpers operate on a discretized aggregate PMF:

from lossmodels.aggregate import (
    var, tvar, stop_loss, lev,                       # on a sample
    var_from_pmf, tvar_from_pmf, stop_loss_from_pmf,  # on an aggregate PMF
)

losses = model.sample(100_000)
print(var(losses, 0.95), tvar(losses, 0.95), stop_loss(losses, 50_000), lev(losses, 50_000))

print(var_from_pmf(agg_pmf, h=100.0, q=0.95))
print(tvar_from_pmf(agg_pmf, h=100.0, q=0.95))
print(stop_loss_from_pmf(agg_pmf, h=100.0, d=50_000.0))

Estimation and model selection

The estimation module provides MLE and method-of-moments fitters, fit diagnostics, goodness-of-fit statistics, and automatic best-fit selection.

from lossmodels import (
    fit_lognormal, fit_poisson,
    fit_best_severity, fit_best_frequency,
    goodness_of_fit, aic, bic,
)

sev_model = fit_lognormal(severity_data)
freq_model = fit_poisson(count_data)

best_sev = fit_best_severity(severity_data, criterion="aic")
best_freq = fit_best_frequency(count_data, criterion="bic")

print(best_sev["best_name"], best_sev["best_model"])
print(best_freq["best_name"])

fit_best_severity / fit_best_frequency return a dict with best_name, best_model, criterion, method, and the full per-candidate results.

• Severity fitters and selection candidates: exponential, gamma, lognormal, pareto, weibull (MLE or method of moments). Also available individually: fit_exponential, fit_gamma, fit_lognormal, fit_pareto, fit_weibull, plus the generic fit_mle.
• Frequency fitters and selection candidates: poisson, negbinomial (fit_poisson, fit_negbinomial).
• Diagnostics and goodness of fit: log_likelihood, aic, bic, ks_statistic, anderson_darling, cramer_von_mises, tail_quantile_table, and goodness_of_fit.

The actuarialpy ecosystem

lossmodels is the distribution-and-aggregation core of a small set of interoperating actuarial packages. Models built here plug into the others through their .sample(size) and .mean() interface:

• risksim — portfolio simulation and aggregate reinsurance program evaluation (layers, contract programs), consuming lossmodels severities and frequencies.
• extremeloss — extreme value theory: peaks-over-threshold / GPD tails, block maxima / GEV, the Hill estimator, and threshold diagnostics. This is the home of the extreme-value distributions, which is why they are not duplicated in lossmodels.

Examples

The repository includes runnable example scripts, including:

• fit_and_simulate.py, fit_and_compare_models.py
• panjer_vs_simulation.py, panjer_vs_fft_vs_simulation.py
• deductible_example.py, limit_example.py, layer_example.py
• stop_loss_example.py

Testing

Run the test suite with:

pytest -q

Run only the fast tests with:

pytest -q -m "not slow"

License

MIT License