survival: implement initial parametric AFT model family

[TheHiddenObserver]

Context

statgpu currently provides semiparametric Cox models but no maintained parametric accelerated-failure-time family. AFT models are the next bounded survival extension after the Cox Phase-1 baseline and complement, rather than replace, proportional-hazards analysis.

Roadmap: PR #89. Nonparametric Kaplan-Meier/Nelson-Aalen work is tracked in #94.

Goal

Implement an initial three-backend AFT framework with Weibull, log-normal, and log-logistic distributions, explicit parameterization, model-based inference, formula support, and survival prediction functions.

Design requirements

Before implementation, document:

• the location/scale and acceleration-factor parameterization;
• coefficient sign convention;
• distribution-specific scale/shape mappings;
• whether a common AFTRegression interface or distribution-specific classes are public;
• intercept and ancillary/scale parameter treatment;
• result and summary object shape;
• sample-weight semantics;
• censoring and entry support;
• optimizer and convergence contract;
• relationship to the existing loss/solver framework.

Do not claim alignment with R survreg or lifelines without an explicit mapping of their coefficient, scale, and shape conventions.

Initial statistical scope

Distributions:

• Weibull;
• log-normal;
• log-logistic.

Data:

• right-censored (time, event) observations;
• positive finite durations;
• optional analytic sample weights only if their likelihood interpretation is fixed and externally validated;
• delayed entry/start-stop may be deferred unless fully supported by the selected likelihood.

Outputs:

• coefficients and ancillary parameters;
• log-likelihood;
• convergence diagnostics;
• model-based covariance, standard errors, test statistics, p-values, and confidence intervals;
• AIC/BIC with a documented parameter count;
• survival, CDF, hazard, cumulative hazard, and quantile predictions;
• conditional median/quantile survival where defined.

Backend contract

• NumPy CPU;
• CuPy CUDA;
• Torch CUDA;
• backend-native likelihood, gradient, Hessian or Hessian-vector products, and prediction;
• explicit device requests must not silently fall back;
• dtype and numerical-stability rules must be tested;
• known domain failures during line search may be handled narrowly, while unrelated runtime/device/OOM errors remain fatal.

Formula contract

• formula/dataframe and array paths must produce aligned design matrices;
• intercept behavior must be explicit;
• categorical terms, interactions, transforms, missing-data row alignment, and prediction column ordering must follow current formula rules;
• ancillary/scale formulas are out of scope unless separately designed.

External alignment

Use matched comparisons against:

• R survival::survreg;
• lifelines Weibull/LogNormal/LogLogistic AFT fitters;
• direct likelihood or simulation checks for parameter mappings.

Compare:

• coefficients after parameter mapping;
• scale/shape parameters;
• log-likelihood;
• covariance/standard errors;
• predicted survival and quantiles;
• AIC/BIC where definitions align.

Non-goals

• no generalized gamma or spline-based flexible parametric survival in the first issue;
• no frailty/random effects;
• no Fine-Gray or multi-state model;
• no time-varying coefficients;
• no survival forest/boosting;
• no nonconvex penalties in the first phase.

Required validation

• exact likelihood/gradient checks on small synthetic data;
• finite-difference or automatic-differentiation gradient/Hessian checks where appropriate;
• uncensored equivalence to the corresponding log-duration regression where applicable;
• censored simulation recovery;
• degenerate and invalid-input tests;
• formula/array parity;
• NumPy/CuPy/Torch parity;
• external alignment with explicit convention mappings;
• physical CuPy and Torch GPU validation;
• EN/CN model documentation and examples;
• benchmark artifact for any performance claim.

Acceptance criteria

• The common AFT parameterization and public API are documented before final implementation.
• Weibull, log-normal, and log-logistic right-censored likelihoods are implemented.
• Fit, summary, likelihood diagnostics, AIC/BIC, and prediction functions are available.
• Model-based covariance and inference pass strict alignment or documented tolerance gates.
• Formula and array APIs agree after row/column alignment.
• NumPy, CuPy, and Torch paths pass parity tests without silent fallback.
• R/lifelines parameter mappings are explicit and externally validated.
• Complete CPU, compatibility, and physical-GPU gates pass.
• English and Chinese documentation are synchronized.

PR decomposition

Prefer staged PRs:

1. common likelihood/result framework plus one distribution;
2. remaining two distributions and external alignment;
3. formula/prediction/documentation and benchmark completion if not already included.