extremeloss

Extreme value theory, tail-risk estimation, and rare-event diagnostics.

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Overview

extremeloss focuses on the part of a loss distribution that is hardest to estimate well: the far tail. It provides peaks-over-threshold and block-maxima workflows, generalized-Pareto and generalized-extreme-value fits, tail-index estimators, tail risk measures (VaR/TVaR) with uncertainty, importance-sampling and bootstrap machinery, and threshold diagnostics — all returning rich result objects you can summarize, plot, and feed back into the rest of a modeling pipeline.

It is built to sit alongside lossmodels (loss distributions and aggregate models) and risksim (portfolio and contract simulation): it can fit a tail directly from a lossmodels object, splice a fitted tail onto a body distribution, or analyze a risksim simulation result. Its only hard dependencies are numpy and scipy.

Highlights

• Peaks-over-threshold (POT) / GPD — extract exceedances, choose a threshold, fit a generalized Pareto distribution, and read off tail probabilities, VaR, TVaR, and return levels.
• Block maxima / GEV — block a series, fit a generalized extreme value distribution, and compute return levels.
• Tail-index estimators — Hill and Pickands estimators, with a Hill curve.
• Threshold diagnostics — mean-excess analysis and a threshold-stability scan.
• Tail risk with uncertainty — empirical and model-based VaR/TVaR/tail probabilities returned as estimates with confidence intervals.
• Variance reduction — importance-sampling estimators with weight diagnostics and effective sample size.
• Bootstrap — resampling uncertainty for tail probabilities, VaR, TVaR, and arbitrary scalar statistics.
• Integration — fit tails from lossmodels objects, splice GPD tails onto bodies, and summarize risksim simulations.

Installation

pip install extremeloss

From source:

pip install -e .

Requires Python >=3.10 with numpy and scipy. The integration helpers use lossmodels / risksim objects when present, and the optional plotting helpers require matplotlib.

Package structure

extremeloss/
├── evt/             # GPD, POT, block-maxima/GEV, tail-index, thresholds
├── estimation/      # empirical, conditional-MC, and importance-sampling estimators
├── analytics/       # return periods/levels, summaries, diagnostics
├── utils/           # bootstrap and validation helpers
├── integration.py   # lossmodels / risksim interop and splicing
├── results.py       # GPDFit, GEVFit, GPDTail, TailEstimateResult, BootstrapResult, ThresholdScan
├── protocols.py     # SupportsSample / SupportsLosses / SupportsSimulationResult
└── plotting.py      # optional matplotlib diagnostics

Quick start

import numpy as np
from extremeloss import fit_pot

losses = np.random.default_rng(0).pareto(2.5, 50_000) * 1000  # heavy-tailed sample

fit = fit_pot(losses, threshold=np.quantile(losses, 0.95))   # -> GPDFit
print("shape xi      :", fit.xi)
print("scale beta    :", fit.beta)
print("99.5% VaR     :", fit.var(0.995))
print("99.5% TVaR    :", fit.tvar(0.995))
print("P(loss > 50k) :", fit.tail_probability(50_000))
print("100-obs return level:", fit.return_level(100))
print(fit.summary())

Peaks-over-threshold and the GPD

The POT workflow: pick a threshold (guided by diagnostics below), keep the exceedances, and fit a generalized Pareto distribution to them.

from extremeloss import extract_exceedances, fit_gpd, fit_pot, gpd_var, gpd_tvar

excesses = extract_exceedances(losses, threshold=10_000)   # losses above the threshold
fit = fit_gpd(excesses, threshold=10_000)                  # fit to excesses, or:
fit = fit_pot(losses, threshold=10_000)                    # fit directly from the data

# functional forms are available when you already have the parameters
v = gpd_var(0.995, threshold=10_000, xi=fit.xi, beta=fit.beta,
            exceedance_fraction=fit.exceedance_fraction)

A GPDFit carries threshold, xi, beta, exceedance_fraction, and n_exceedances, and exposes var(p), tvar(p), tail_probability(x), return_level(period), and summary().

Block maxima and the GEV

from extremeloss import make_blocks, fit_block_maxima, fit_gev, block_return_level

maxima = make_blocks(losses, block_size=250)        # block maxima
gev = fit_block_maxima(losses, block_size=250)      # or fit_gev(maxima)
print(gev.xi, gev.loc, gev.scale)
print("100-block return level:", gev.return_level(100))

A GEVFit carries xi, loc, scale, and n_blocks, and exposes return_level(period), cdf(x), and summary().

Threshold diagnostics

Choosing the POT threshold is the crux of a good tail fit. Mean-excess and threshold-stability scans help:

from extremeloss import mean_excess, threshold_diagnostic_table

grid = np.quantile(losses, [0.90, 0.95, 0.975, 0.99])
me = mean_excess(losses, grid)                       # mean excess at each threshold
scan = threshold_diagnostic_table(losses, grid)      # -> ThresholdScan
print(scan.thresholds, scan.xi, scan.beta, scan.n_exceedances)

A ThresholdScan exposes thresholds, mean_excess, xi, beta, n_exceedances, and to_dict(). Look for the threshold above which the shape xi and the mean excess stabilize.

Tail-index estimators

from extremeloss import hill_estimator, hill_curve, pickands_estimator

hill_estimator(losses, k=500)     # Hill tail-index estimate using the top k order statistics
pickands_estimator(losses, k=500)
curve = hill_curve(losses)        # Hill estimate across a grid of k (for a Hill plot)

Return periods and levels

from extremeloss import return_period, return_level, block_return_level

return_period(0.01)             # expected waiting time for a 1% exceedance
return_level(100, fit)          # the level exceeded once per 100 observations (GPDFit)
block_return_level(100, gev)    # the level exceeded once per 100 blocks (GEVFit)

Tail risk estimation with uncertainty

Empirical estimators are one-liners; the model-based estimators return a TailEstimateResult with a confidence interval and standard error.

from extremeloss import empirical_var, empirical_tvar, estimate_var, estimate_tvar, estimate_tail_probability

empirical_var(losses, 0.99)
empirical_tvar(losses, 0.99)

res = estimate_var(losses, 0.99)          # -> TailEstimateResult
print(res.estimate, res.ci, res.stderr)
print(res.summary())

estimate_tvar(losses, 0.99)
estimate_tail_probability(losses, threshold=100_000)

Conditional Monte Carlo summaries are available when you already have per-scenario conditional probabilities or tail expectations (estimate_tail_probability_cmc, estimate_tvar_cmc).

Importance sampling

When the event of interest is rare, importance sampling estimates it far more efficiently than crude Monte Carlo. Supply the losses drawn from a proposal distribution and their importance weights:

from extremeloss import (
    estimate_var_is, estimate_tvar_is, estimate_tail_probability_is,
    importance_sampling_diagnostics, effective_sample_size, stabilize_weights,
)

# losses and weights come from your proposal sampler
res = estimate_tail_probability_is(losses, weights, threshold=1_000_000)
print(res.estimate, res.ci, res.effective_n)

estimate_var_is(losses, weights, 0.999)
estimate_tvar_is(losses, weights, 0.999)

print(importance_sampling_diagnostics(weights))   # weight quality metrics
print(effective_sample_size(weights))
w = stabilize_weights(weights, clip_quantile=0.999)  # tame extreme weights

log_importance_weights(log_target_density, log_proposal_density) builds normalized weights directly from log-densities, and estimate_mean_is / estimate_exceedance_curve_is / estimate_var_tvar_is cover means, full exceedance curves, and joint VaR/TVaR.

Bootstrap uncertainty

from extremeloss import bootstrap_var, bootstrap_tvar, bootstrap_tail_probability, bootstrap_statistic

bv = bootstrap_var(losses, 0.99, n_resamples=1000)   # -> BootstrapResult
print(bv.estimate, bv.ci, bv.stderr)

bootstrap_tvar(losses, 0.99)
bootstrap_tail_probability(losses, threshold=100_000)

# wrap any scalar statistic
bootstrap_statistic(losses, np.median, n_resamples=1000)

A BootstrapResult carries estimate, bootstrap_estimates, ci, stderr, and summary().

The GPD tail object and splicing

GPDTail turns a fitted tail into a standalone, sampleable severity supported above the threshold (sample, cdf, quantile, mean, variance). Because its cdf(threshold) = 0, it satisfies the splicing contract used by lossmodels.SplicedSeverity, letting you attach an EVT tail to a body distribution:

from extremeloss import fit_pot, GPDTail, splice_gpd_tail, fit_spliced_gpd
from lossmodels import Gamma, SplicedSeverity

fit = fit_pot(losses, threshold=10_000)
tail = GPDTail.from_fit(fit)             # a severity for the tail above 10k

# splice an EVT tail onto a lossmodels body, two equivalent ways:
spliced = splice_gpd_tail(Gamma(2.0, 4_000), fit)            # via extremeloss helper
spliced = fit_spliced_gpd(Gamma(2.0, 4_000), losses, threshold=10_000)

# or assemble it yourself with lossmodels:
body = Gamma(2.0, 4_000)
u = tail.threshold
spliced = SplicedSeverity(body=body, tail=tail, threshold=u, weight=body.cdf(u))

The spliced object is a full lossmodels severity, so it drops straight back into a collective-risk model or a risksim portfolio.

Integrating with loss models and simulations

extremeloss reads directly from lossmodels and risksim objects:

Helper	Purpose
sample_lossmodel(model, size)	draw a sample from any lossmodels model
fit_pot_from_lossmodel(model, size, threshold)	sample a model and fit a GPD tail in one step
losses_from_risksim(result, view)	pull a loss vector (gross / ceded / retained) from a risksim SimulationResult
tail_summary_from_risksim(result, ...)	a tail summary of a risksim simulation
component_tail_metrics(result, q, ...)	per-component tail metrics from a simulation
layer_tail_metrics(result, q, ...)	per-layer tail metrics from a simulation

from extremeloss import fit_pot_from_lossmodel, losses_from_risksim, tail_summary_from_risksim
from lossmodels import ParetoII

fit = fit_pot_from_lossmodel(ParetoII(2.5, 1000), size=40_000, threshold=1_500)

# after running a risksim Portfolio.simulate(...) -> result:
# net_losses = losses_from_risksim(result, view="retained")
# summary = tail_summary_from_risksim(result, quantiles=(0.95, 0.99, 0.995))

Plotting (optional)

With matplotlib installed, the extremeloss.plotting module offers diagnostic plots — plot_mean_excess(losses, thresholds), plot_hill_curve(losses), and plot_exceedance_curve(losses, thresholds) (each accepts an optional ax).

Testing

pytest -q

License

MIT License