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Reliability Estimation in Series Systems: Maximum Likelihood Techniques for Right-Censored and Masked Failure Data

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reliability statistics weibull-distribution maximum-likelihood bootstrap series-systems censored-data

My master’s thesis in mathematics investigating maximum likelihood techniques to estimate component reliability from masked failure data in series systems.

The Challenge

Quantifying the reliability of individual components in a series system is challenging when:

  • Only system-level failure data is observable
  • The exact component cause of failure is masked (ambiguous)
  • System lifetimes are right-censored
  • Sample sizes are small

A series system fails when any component fails, making it critical to understand which components are weakest.

Key Contributions

1. Likelihood Model for Masked Data

Developed a likelihood model that handles two types of masking:

Right-Censoring: System observed until time \(\tau\), but may not have failed

\[ S_i = \min\{\tau_i, T_i\} \]

\[ \delta_i = \mathbb{1}_{T_i < \tau_i} \]

Component Cause Masking: When system fails, we observe a candidate set \(\mathcal{C}_i\) containing the failed component, but cannot pinpoint the exact cause.

Three Key Conditions (satisfied in many industrial scenarios):

  1. Candidate set always contains the failed component: \(\Pr\{K_i \in \mathcal{C}_i\} = 1\)
  2. Masking probability is the same for all components in the candidate set
  3. Masking probabilities don’t depend on system parameters \(\theta\)

Under these conditions, the likelihood contribution simplifies to:

\[ L_i(\theta) \propto \left[\prod_{j=1}^m R_j(s_i; \theta_j)\right] \times \left[\sum_{j \in \mathcal{C}_i} h_j(s_i; \theta_j)\right]^{\delta_i} \]

where \(R_j\) is the reliability function and \(h_j\) is the hazard function of component \(j\).

2. Weibull Series Systems

For components with Weibull lifetimes:

  • Component \(j\): \(T_{ij} \sim \text{Weibull}(k_j, \lambda_j)\)
  • Shape parameter \(k_j\) determines failure behavior:
    • \(k < 1\): infant mortality
    • \(k = 1\): random failures (exponential)
    • \(k > 1\): wear-out failures

System reliability when all components have Weibull lifetimes:

\[ R_{T_i}(t; \theta) = \exp\left\{-\sum_{j=1}^m \left(\frac{t}{\lambda_j}\right)^{k_j}\right\} \]

The hazard function is additive:

\[ h_{T_i}(t; \theta) = \sum_{j=1}^m \frac{k_j}{\lambda_j}\left(\frac{t}{\lambda_j}\right)^{k_j-1} \]

3. Extensive Simulation Studies

Validated MLE performance under varying conditions:

Scenario 1: Right-Censoring Impact (\(q = 60\%\) to \(100\%\))

  • Finding: Scale parameters showed positive bias with censoring
  • Finding: Shape parameters more sensitive than scale parameters
  • Finding: Most reliable component (component 1) most affected by censoring
  • Finding: Convergence rate \(>95\%\) for \(q \geq 0.7\)

Scenario 2: Masking Probability (\(p = 10\%\) to \(70\%\))

  • Finding: Scale parameter CIs correctly specified up to \(p=0.7\) (\(>90\%\) coverage)
  • Finding: Shape parameter CIs correctly specified up to \(p=0.4\)
  • Finding: Convergence rate \(>95\%\) for \(p \leq 0.4\)
  • Finding: Bias increases with masking probability

Scenario 3: Sample Size (\(n = 50\) to \(500\))

  • Finding: At \(n=250\), estimator essentially unbiased despite moderate censoring and masking
  • Finding: Convergence rate \(>95\%\) for \(n \geq 100\)
  • Finding: Precision improves rapidly with sample size
  • Finding: Small samples (\(n<100\)) require caution

4. BCa Bootstrap Confidence Intervals

Bias-Corrected and Accelerated (BCa) bootstrap provides:

  • Bias correction: Adjusts for bias in bootstrap distribution
  • Acceleration: Accounts for parameter-dependent distribution shape

Key Results:

  • Good coverage probability (\(>90\%\)) even for small samples
  • Scale parameters better calibrated than shape parameters
  • Coverage approaches nominal \(95\%\) as sample size increases
  • CIs neither too wide nor too narrow

Performance Summary

Despite significant masking and censoring, the MLE demonstrates:

✓ Good accuracy for moderate sample sizes (\(n \geq 100\)) ✓ Reasonable precision with well-specified confidence intervals ✓ High convergence rates (\(>95\%\)) under realistic conditions ✓ Robust performance for well-designed series systems

Challenges identified:

  • Shape parameters harder to estimate than scale parameters
  • Most reliable components require more data
  • Small samples (\(n<100\)) need careful interpretation
  • Severe masking/censoring reduces reliability

Practical Applications

This methodology applies to:

  • Industrial reliability testing with incomplete failure diagnosis
  • Electronic system reliability with board-level diagnostics
  • Mechanical systems where root cause analysis is expensive
  • Any series system with Weibull component lifetimes

Technical Implementation

Full R implementation available: wei.series.md.c1.c2.c3

Features:

  • Analytical score function for efficient optimization
  • L-BFGS-B solver with bound constraints
  • BCa bootstrap confidence intervals
  • Bernoulli masking model for simulations
  • Complete simulation framework

Full Thesis

View complete PDF (40 pages)

Abstract: This paper investigates maximum likelihood techniques to estimate component reliability from masked failure data in series systems. A likelihood model accounts for right-censoring and candidate sets indicative of masked failure causes. Extensive simulation studies assess the accuracy and precision of maximum likelihood estimates under varying sample size, masking probability, and right-censoring time for components with Weibull lifetimes.

For more research and projects, see my research page and publications.

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