API Reference

Distributions

Parametric, discretized and sampled representations.

distributions.Lognormal: Lognormal distribution: ln X ~ Normal(meanlog, sdlog**2).
distributions.Gamma: Gamma distribution with shape alpha and scale theta: mean
distributions.Tweedie: Tweedie distribution with mean mu, dispersion phi and power
distributions.Weibull: Weibull distribution with shape k and scale lam:
distributions.Loglogistic: Loglogistic (Fisk) distribution with shape alpha and scale
distributions.Mixture: A finite mixture of severities: component i with probability
distributions.Custom: A loss severity defined by your own distribution function: the slow
distributions.to_json(): A distribution as a JSON document: the family and the parameters its
distributions.from_json(): A distribution from a document written by to_json, as the class of
distributions.Grid: A distribution on the points 0, step, 2*step, ...: the discretized
distributions.DiscretizationReport: How a distribution was discretized, and the error that introduced.
distributions.Sampled: A distribution known only through equally weighted draws.

Large-loss severities

The Pareto family for layer and treaty pricing, with maximum likelihood fits.

distributions.Pareto: Single-parameter Pareto: P(X > x) = (t / x) ** alpha for x >= t,
distributions.PiecewisePareto: Piecewise Pareto: alpha alpha[k] above threshold t[k], the
distributions.LogAffinePareto: Log-affine local Pareto: the local alpha
distributions.GeneralizedPareto: Generalized Pareto severity with a location (Riegel’s parameterization
distributions.local_pareto_to_piecewise(): Converts the local Pareto distribution with local alpha alpha(x)

Claim counts

Frequency distributions for frequency-severity aggregation.

distributions.Poisson: Poisson claim counts with mean lam.
distributions.NegativeBinomial: Negative binomial claim counts: mean r * beta, variance
distributions.Binomial: Binomial claim counts: n risks, each claiming with probability
distributions.claim_count(): The claim count with this mean and dispersion Var[N] / E[N]:

Predictive distributions

The joint result every model returns.

distributions.PredictiveDistribution: The joint result every model returns: draws for each simulation (row)

Aggregate loss

Compound distributions on a grid, and simulated years of losses.

aggregate.panjer(): Aggregate loss S = X_1 + ... + X_N by Panjer’s recursion.
aggregate.fft(): Aggregate loss S = X_1 + ... + X_N by fast Fourier transform.
aggregate.CompoundReport: What a compound calculation produced and the error it introduced.
aggregate.simulate_events(): Simulates n_sims years of claims: a count from frequency, then that
aggregate.EventSet: Simulated years of individual losses, for applying per-loss terms such

Reinsurance

Excess-of-loss layers and towers, applied to simulated losses or exactly on the grid.

reinsurance.Layer: A per-occurrence excess-of-loss layer: limit xs attachment on each
reinsurance.Tower: A reinsurance programme: layers in inuring stages.
reinsurance.TowerGrids: A tower’s annual distributions on the grid, from Tower.on_grid.

Models

Terms and design matrices, GLMs and GAMs, metrics, resampling and MCMC diagnostics.

models.Terms: The terms of a model: an intercept, numeric columns and factors.
models.Coding: Terms with factor levels learned from training data, from
models.Design: A design matrix with an offset and prior weights.
models.Glm: A generalized linear model, fitted by IRLS.
models.GlmFit: A fitted GLM, from Glm.fit.
models.BayesGlm: A Bayesian GLM sampled with NUTS (nuts-rs, the Rust core of nutpie).
models.BayesGlmFit: A sampled Bayesian GLM, from BayesGlm.fit.
models.ElasticNet: An elastic-net GLM: the lasso (alpha=1), ridge (alpha=0) and
models.ElasticNetFit: A fitted elastic net, from ElasticNet.fit or ElasticNet.path.
models.CvPath: Cross-validated scores along an elastic-net path, from
models.Gam: A generalized additive model: a Glm plus P-spline smooths of
models.GamFit: A fitted GAM, from Gam.fit.
models.deviance(): Deviance sum w d(y, mu) of a family.
models.gini(): Gini index of the ordered Lorenz curve.
models.lift(): Lift table: rows sorted by predicted rate, cut into bands of about
models.crps(): Continuous ranked probability score of equally likely draws for an
models.log_score(): Mean log score -(1/n) sum log f(y_i) of the outcomes under the
models.pit(): Probability integral transform of each outcome under the family’s
models.pit_from_draws(): The PIT of y under the empirical distribution of draws,
models.pit_histogram(): Counts of PIT values in bins equal-width bins of [0, 1].
models.ks_uniform(): Kolmogorov-Smirnov distance from the uniform on [0, 1]; about
models.k_fold(): k-fold splits of n rows, shuffled with seed.
models.group_k_fold(): Grouped k-fold splits: each group’s rows stay in one fold.
models.time_ordered(): Time-ordered splits: for each of the last n_test periods, train on
models.mcmc_diagnostics(): MCMC diagnostics of chains of draws (Vehtari et al. 2021, as R’s
models.elpd_loo(): Leave-one-out cross-validation by Pareto-smoothed importance sampling
models.stacking_weights(): Stacking weights from pointwise held-out log predictive densities
models.pseudo_bma_weights(): Pseudo-BMA weights, w_k proportional to exp(elpd_k); with
models.BayesStacking: Bayesian stacking: a posterior for the stacking weights, with a
models.HierarchicalStacking: Hierarchical stacking (Yao, Pirš, Vehtari and Gelman, 2022): model
models.StackingFit: Posterior stacking weights, from BayesStacking.fit or
models.elpd_waic(): WAIC from pointwise log-likelihood draws: lppd less the variance of
models.lppd(): In-sample log pointwise predictive density,
models.Elpd: An ELPD estimate from elpd_loo or elpd_waic.
models.deviance_score(): A score for cross_validate: the family’s mean deviance on the
models.cross_validate(): Fits model on each split’s training rows and scores it on the
models.grid_search(): Scores every candidate by cross_validate on the same splits and
models.random_search(): Draws n candidates with draw(rng) from random.Random(seed)
models.log_uniform(): A draw log-uniform between low and high, for learning rates
models.SearchResult: The result of grid_search or random_search.
models.compare(): Fits every model on each split’s training rows and scores its test
models.Comparison: The result of compare: every model’s score on every metric and
models.actual_vs_expected(): Actual against expected by period for a stored model’s predictions on
models.simulate_from_means(): Joint predictive draws from fitted means, for engines that give only

Gradient boosting

LightGBM and XGBoost behind the model protocol, with joint predictive draws.

boosting.Booster: Gradient-boosted trees for a response family, through LightGBM or
boosting.BoosterFit: A fitted Booster: means and joint predictive draws.

Pricing

The collective model, layer rating, reinsurance tower matching, and risk-loaded prices from simulated losses.

pricing.CollectiveModel: The collective risk model: a claim count and a severity, with layer
pricing.ilf(): Increased limit factor LEV(limit) / LEV(basic_limit).
pricing.loss_elimination_ratio(): Loss elimination ratio of a deductible, LEV(deductible) / E[X].
pricing.pareto_extrapolation(): Expected loss of layer to per unit of expected loss of layer
pricing.alpha_between_layers(): The Pareto alpha at which two layers have the given expected losses.
pricing.alpha_between_frequency_and_layer(): The Pareto alpha at which frequency losses a year above
pricing.alpha_between_frequencies(): The Pareto alpha between two excess frequencies.
pricing.match_tower(): Matches a tower of contiguous layers, the last unlimited, with one
pricing.fit_pml_curve(): The model through the points of a PML curve: amounts[j] is
pricing.fit_references(): A model that reproduces every reference: expected layer losses (which
pricing.TowerModel: A frequency and a piecewise Pareto severity that reproduce a tower,
pricing.price(): Risk-loaded price of a cover from its simulated losses.
pricing.price_portfolio(): Prices a portfolio and allocates the price to its components.
pricing.Price: The risk-loaded price of a cover, or of one component’s share of a
pricing.PortfolioPrice: Prices of a portfolio’s components and of the portfolio as a whole.
pricing.Mbbefd: The MBBEFD exposure curve and destruction-rate distribution (Bernegger,
pricing.TabulatedCurve: A tabulated exposure curve: points (x, G(x)) from (0, 0) to
pricing.RiskProfile: A risk profile for property per-risk business: bands of sum insured,
pricing.severity_exposure_curve(): The exposure curve of a severity capped at the maximum possible loss

Reserving

Loss triangles, the chain ladder with tail factors, Mack’s model with its one-year view, the expected-loss methods, the ODP bootstrap and Clark’s growth curves.

reserving.Triangle: A loss triangle with four axes: index (segment), column (measure), origin
reserving.ChainLadder: The chain-ladder method: each origin’s latest value projected to
reserving.ChainLadderFit: A fitted chain-ladder projection of every segment of a triangle column.
reserving.TailConstant: A given tail factor, as chainladder-python’s TailConstant.
reserving.TailCurve: A curve fitted to the estimated factors and extrapolated, as
reserving.TailBondy: The Bondy tail, as chainladder-python’s TailBondy.
reserving.TailLogLinear: R ChainLadder’s tail = TRUE rule (its tailfactor function).
reserving.Mack: Mack’s distribution-free chain ladder: the chain-ladder projection plus
reserving.MackFit: A fitted Mack model of every segment: the chain-ladder fields, plus
reserving.ClaimsDevelopmentResult: Merz and Wüthrich’s (2008) one-year view of a Mack fit: standard errors
reserving.ExpectedLoss: The expected loss ratio method: each origin’s ultimate is apriori
reserving.BornhuetterFerguson: The Bornhuetter–Ferguson method: each origin’s latest value plus the
reserving.Benktander: The Benktander (iterated Bornhuetter–Ferguson) method: starting from
reserving.CapeCod: The Cape Cod (Stanard–Bühlmann) method: Bornhuetter–Ferguson with each
reserving.ExpectedLossFit: A fitted expected-loss method (ExpectedLoss,
reserving.CapeCodFit: A fitted Cape Cod of every segment of a triangle column: the fields of
reserving.OdpBootstrap: Over-dispersed Poisson bootstrap of the chain ladder (England and
reserving.OdpBootstrapFit: A fitted ODP bootstrap of every segment.
reserving.MackBootstrap: Mack’s bootstrap for the lifetime and one-year views (England, Verrall
reserving.MackBootstrapFit: A fitted bootstrap of Mack’s model, the lifetime view, of every
reserving.OneYearFit: The simulated one-year view of every segment, from
reserving.ClarkLdf: Clark’s LDF method (Clark 2003), as R ChainLadder’s ClarkLDF: each
reserving.ClarkCapeCod: Clark’s Cape Cod method (Clark 2003), as R ChainLadder’s
reserving.ClarkFit: A fitted Clark LDF or Cape Cod model of every segment of a triangle

Risk measures

Distortion risk measures, and capital allocation to components.

risk.Distortion: A distortion risk measure: rho(X) = integral of g(S(x)) dx for a
risk.allocate(): Allocates a distortion risk measure of the total to the components.
risk.capital(): Allocates the distortion risk measure of a portfolio’s total to its
risk.entropic(): Entropic risk measure (1 / theta) log E[exp(theta X)]: the certainty
risk.esscher(): Esscher premium E[X exp(h X)] / E[exp(h X)]: the mean after tilting
risk.marginal_expected_shortfall(): Marginal expected shortfall of each component at level p: its mean
risk.covar(): CoVaR of a component: the total’s VaR at level q over the
risk.esscher_allocation(): Esscher allocation: each component’s mean under the Esscher transform
risk.Allocation: Capital allocation of a distortion risk measure, from capital.

Dependence

Copulas, and reordering existing draws to a target correlation.

risk.GaussianCopula: The Gaussian copula with correlation matrix correlation.
risk.StudentTCopula: The Student t copula with correlation matrix correlation and nu
risk.ArchimedeanCopula: An exchangeable Archimedean copula: Clayton, Gumbel, Frank or Joe.
risk.simulate(): Simulates marginals joined by a copula.
risk.iman_conover(): Reorders each component’s draws to a target correlation (Iman-Conover).

Extreme value tails

Generalized Pareto tails over a threshold, beyond the draws.

risk.Gpd: The generalized Pareto distribution, as SciPy’s
risk.PotTail: A peaks-over-threshold tail: draws above a threshold modelled by a
risk.mean_excess(): The empirical mean-excess function e(u) = E[X - u | X > u] at each
risk.hill(): Hill estimates of the tail index xi (1 / alpha) from the k