Consequence Analysis for Weibull Series Systems
When a reliability engineer simplifies a series system model by assuming all components share a common Weibull shape parameter, how much accuracy is lost -- and can the data even tell the difference? This paper shows these two questions have a quantitatively aligned answer.
The common-shape model's MTTF prediction bias and the likelihood ratio test's detection power are aligned as functions of shape heterogeneity:
| Shape CV | MTTF Bias | LRT Power (n=500) | Recommendation |
|---|---|---|---|
| < 5% | < 0.2% | < 7% | Use reduced model confidently |
| 5--14% | 0.2--0.9% | 7--37% | Safe zone: bias negligible, test lacks power |
| 14--21% | 0.9--1.5% | 37--97% | Transition: test detects meaningful bias |
| > 27% | > 2.9% | > 99% | Use full model |
The test lacks power precisely where the consequence of using the wrong model is negligible, and has high power precisely where bias warrants switching models.
A bias-variance decomposition reveals that the full model has lower MTTF variance than the reduced model even when the reduced model is correctly specified (CV = 0). At n = 100: full variance = 348, reduced variance = 417. The constraint k_1 = ... = k_m forces all shape estimation error into one degree of freedom, amplifying its propagation through the nonlinear MTTF integral.
An LRT-based adaptive procedure achieves RMSE within 2.5% of the always-full strategy at n >= 500, while selecting the simpler model over 90% of the time when appropriate.
| Criterion | Type I Error | Behavior |
|---|---|---|
| LRT (alpha = 0.05) | 4.6--6.8% | Well-calibrated |
| AIC | 8.2--12.4% | Liberal (~2x nominal) |
| BIC | 0--0.2% | Over-conservative |
.
├── paper/ # LaTeX source and figures
│ ├── paper.tex # Main manuscript
│ ├── refs.bib # Bibliography
│ ├── Makefile # Build automation
│ └── image/ # Figures (PDF)
├── qrei/ # QREI submission variant
│ ├── manuscript.tex # Journal-formatted manuscript
│ └── cover-letter.tex # Cover letter
├── results/ # Simulation code and data
│ ├── consequence/ # Consequence analysis (MTTF bias, MSE)
│ │ └── figure/ # Including bias-variance decomposition
│ ├── adaptive/ # Adaptive model selection
│ ├── lrt/ # Likelihood ratio test simulations
│ │ ├── divergence/ # Type I error and power vs shape CV
│ │ ├── vary_m/ # Effect of number of components (m = 2-8)
│ │ ├── vary_p/ # Effect of masking probability (p = 0.05-0.70)
│ │ ├── vary_q/ # Effect of censoring level (q = 0.50-1.00)
│ │ └── nomasking/ # Ideal case baseline (p=0, q=1)
│ ├── 5_system_scale3/ # Scale parameter sensitivity
│ ├── 5_system_shape3/ # Shape parameter sensitivity
│ └── sim_utils.R # Shared vectorized simulation utilities
├── docs/ # GitHub Pages (paper PDF)
└── CLAUDE.md # Development guidance
R scripts (Monte Carlo) → CSV → Python analysis → PDF figures → LaTeX paper
Each experiment in results/ contains an R script that runs simulations, a CSV of raw results, and a figure/ subdirectory with a Python analysis script that produces publication-quality PDFs.
cd paper
latexmk -pdf paper.texOr with Make:
cd paper
make # build paper.pdf
make docs # copy PDF to docs/ for GitHub Pageswei.series.md.c1.c2.c3-- Weibull series system with masked dataalgebraic.mle-- Maximum likelihood estimation utilitiesalgebraic.dist-- Algebraic distribution functionsmd.tools-- Masked data toolstidyverse,ggplot2,parallel,boot
matplotlib,seaborn,pandas,numpy
Standard distribution with amsmath, amsthm, graphicx, natbib, hyperref
- Towell (2023). Reliability Estimation in Series Systems with Masked Data -- the C1-C2-C3 likelihood framework
wei.series.md.c1.c2.c3-- R package implementing the parametric models
@article{towell2025consequence,
title={When Does Model Simplification Matter? Consequence Analysis for Weibull Series Systems},
author={Towell, Alexander},
year={2025},
note={Preprint},
url={https://github.com/queelius/weibull-series-consequence}
}MIT License -- see the LICENSE file for details.
Alexander Towell ORCID: 0000-0001-6443-9897 lex@metafunctor.com