mdrelax extends my work on series system reliability by handling cases where the standard masking assumptions break down.

Background: The C1-C2-C3 Framework

My master’s thesis developed maximum likelihood techniques for series systems with masked failure data. The standard framework assumes three conditions:

• C1: The failed component is always in the candidate set
• C2: Non-informative masking (uniform probability within candidate set)
• C3: Masking mechanism is independent of system parameters

When these hold, the masking probabilities factor out and you can ignore them for parameter estimation. The expo-masked-fim paper derives closed-form Fisher Information for the exponential case, and maskedcauses implements the general framework.

The Problem

In practice, C2 and C3 are often violated.

Informative masking (C2 violation): Diagnostic tests may be better at identifying certain failure modes than others. A component that fails catastrophically is easier to identify than one that degrades subtly.

Parameter-dependent masking (C3 violation): The masking mechanism might depend on component reliabilities. Components with shorter lifetimes fail more often, so technicians get more practice diagnosing them.

If you pretend C2 and C3 hold when they don’t, your parameter estimates are biased. Sometimes badly.

What mdrelax Does

The package implements likelihood-based inference with relaxed conditions:

library(mdrelax)

# Generate masked data with Bernoulli candidate sets
md <- md_bernoulli_cand_C1_C2_C3(data, p = 0.3)

# Sample candidate sets
md <- md_cand_sampler(md)

# MLE for exponential series system
fit <- md_mle_exp_series_C1_C2_C3(md)

# Fisher information matrix
fim <- md_fim_exp_series_C1_C2_C3(md, params(fit))

Key Features

• Flexible masking models: Bernoulli, rank-based, KL-divergence constrained
• Identifiability analysis: Tools to check when parameters can actually be estimated
• Fisher information: Efficiency analysis under relaxed conditions
• Simulation utilities: Monte Carlo studies for method validation

Relationship to Other Work

This package sits at the end of a progression toward generality:

The progression:

1. Exponential + C1-C2-C3: Closed-form solutions
2. Weibull + C1-C2-C3: Numerical MLE
3. Weibull + relaxed conditions: mdrelax

Each step trades analytical tractability for realism.

When to Use It

Use mdrelax when you suspect:

• Diagnostic accuracy varies by component type
• Masking patterns correlate with component reliabilities
• Standard C1-C2-C3 assumptions are too restrictive for your data

The trade-off is real: relaxed models have more parameters and may need larger samples for reliable estimation. But biased estimates from wrong assumptions aren’t free either.

When can you safely use a simpler model for a series system? I ran extensive simulation studies with likelihood ratio tests to get a quantitative answer.

The Problem

In series system reliability, you estimate component parameters from masked failure data. For Weibull components, that means estimating \(2m\) parameters: shape \(k_j\) and scale \(\lambda_j\) for each of \(m\) components.

But what if the components have similar failure characteristics? A reduced model with homogeneous shape parameters uses only \(m+1\) parameters (one common \(k\) plus \(m\) scales). This roughly halves the parameter count and has a nice property: the system itself becomes Weibull-distributed.

The question is when this simplification is justified.

Key Findings

Robustness of the Reduced Model

For well-designed series systems (components with similar failure characteristics), the result is striking:

The reduced homogeneous-shape model cannot be rejected even with sample sizes approaching 30,000, far larger than anything typically available in practice.

With realistic sample sizes (50 to 500), the likelihood ratio test shows no evidence against the reduced model when components truly have similar shapes. This is strong justification for using the simpler model.

Sharp Boundaries

The paper pins down exactly how much heterogeneity it takes to trigger rejection:

Even modest deviations in a single component’s shape parameter provide evidence against the reduced model. The boundaries are clean.

Practical Guidance

Use the reduced model when:

• Components come from similar manufacturing processes
• Historical data suggests similar wear-out patterns
• Sample sizes are moderate (\(n < 500\))
• You need a quick reliability assessment

Use the full model when:

• Components have fundamentally different failure modes (infant mortality vs wear-out)
• Large samples are available (\(n > 1000\))
• Precise component-level inference is critical
• Preliminary studies suggest model inadequacy

Connection to Related Work

This paper fits into a broader program on masked failure data:

The pieces address different aspects of the same problem:

This is my master’s thesis in mathematics. The problem: you have a series system (fails when any component fails), you can observe system-level failure times, but you often can’t tell which component actually caused the failure. The failure cause is “masked.” On top of that, some systems are still running at the end of the study, so their lifetimes are right-censored. You want to estimate the reliability of individual components from this incomplete data.

The challenge

Estimating component reliability is hard when:

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

A series system fails when any component fails, so disentangling which components are weakest from system-level observations is a non-trivial inference problem.

Likelihood model for masked data

I developed a likelihood model that handles two types of incompleteness.

Right-censoring: the system is observed until time \(\tau\), but may not have failed yet:

Component cause masking: when the system fails, you observe a candidate set \(\mathcal{C}_i\) containing the failed component, but can’t pinpoint the exact cause.

Under three conditions (which hold in many industrial settings), the likelihood contribution simplifies to:

where \(R_j\) is the reliability function and \(h_j\) is the hazard function of component \(j\). The three conditions are: the candidate set always contains the true failed component, masking probability is uniform across components in the candidate set, and masking probabilities don’t depend on the system parameters \(\theta\).

Weibull series systems

I focused on components with Weibull lifetimes: \(T_{ij} \sim \text{Weibull}(k_j, \lambda_j)\). The shape parameter \(k_j\) tells you the failure behavior: \(k < 1\) is infant mortality, \(k = 1\) is random failures (exponential), \(k > 1\) is wear-out.

System reliability when all components are Weibull:

The hazard function is additive:

Simulation studies

I ran extensive simulations varying three factors:

Right-censoring impact (q = 60% to 100%): Scale parameters showed positive bias with censoring. Shape parameters were more sensitive than scale parameters. The most reliable component was most affected by censoring. Convergence rate exceeded 95% for q >= 0.7.

One of the most interesting statistical problems I have encountered is reliability analysis with censored data: situations where you know something didn’t fail, but not when it will fail.

The Censoring Problem

Imagine testing light bulbs. You run them for 1000 hours. Some fail during the test. Others are still working when you stop.

For the survivors, you know:

• They lasted at least 1000 hours
• You do not know their actual lifetime

This is right censoring. The true value lies somewhere to the right of your observation. You have a lower bound, not a measurement.

Why This Matters

Censored data is everywhere:

• Medical studies (patients still alive at study end)
• Engineering tests (components that have not failed)
• Customer retention (users still active)

The naive responses are both wrong. Ignoring censored observations wastes information. Treating them as failures introduces bias. You need a framework that uses the partial information you actually have.

Maximum Likelihood to the Rescue

The solution is maximum likelihood estimation with likelihood contributions that account for censoring:

• Failure observations contribute the probability density \(f(t)\). You observed the exact failure time, so you know the probability of failing at that time.
• Censored observations contribute the survival probability \(S(t)\). You know the unit survived to time \(t\), so its contribution is the probability of surviving at least that long.

The likelihood for the whole sample is:

This lets you extract information from both failed and surviving units. The censored observations pull the estimated reliability upward; the failures pull it downward. Maximum likelihood balances them.

Series Systems Complexity

It gets more interesting with series systems, systems that fail when any component fails. If you observe system failure but do not know which component caused it, you have masked failure data.

This is the problem I am most interested in: extracting component-level reliability from system-level failures when the cause is ambiguous. The masking adds a latent variable, and the likelihood becomes a mixture. You can handle it with EM algorithms or direct optimization, but the combinatorics grow quickly with system size.

This work is laying groundwork for what will become a major focus of my mathematical statistics degree.