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Closed-Form Results for Masked Exponential Series Systems

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statistics fisher information masked data series systems reliability exponential distribution maximum likelihood sufficient statistics

This paper provides complete analytical results for maximum likelihood estimation in series systems with masked failure data under exponential component lifetimes. Unlike numerical approaches, everything here has a closed form.

The Problem

In series systems, the system fails when any component fails. You observe:

  • System failure time \(t\)
  • Candidate set \(C \subseteq \{1,2,\ldots,m\}\) of possible failed components

But you don’t know which component in \(C\) actually caused the failure. This is masked failure data.

Key Contributions

1. Closed-Form Fisher Information Matrix

For exponential masked data with arbitrary masking patterns:

$$I_{ij}(\boldsymbol{\lambda}) = n \cdot \sum_{A \ni i,j} \frac{\hat{\omega}_A}{(\sum_{k \in A} \lambda_k)^2}$$

where \(\hat{\omega}_A\) is the observed frequency of candidate set \(A\). This enables:

  • Direct computation of asymptotic variances
  • Identifiability checking before running MLE
  • Stability analysis of optimization

2. Sufficient Statistics

The mean system lifetime and candidate set frequency vector constitute sufficient statistics, reducing an entire dataset to just \(1 + \binom{m}{w}\) numbers (where \(w\) is the masking width).

This is a major simplification: all statistical information is captured by counting which candidate sets appear and what the average failure time is.

3. Closed-Form MLE for Three-Component Systems

For \(m=3\) components with pairwise masking (\(w=2\)), we derive an explicit closed-form solution to the likelihood equations:

$$\hat{\lambda}_j = \frac{\sum_{A \ni j} \hat{\omega}_A}{\bar{t} \cdot n}$$

This eliminates numerical optimization entirely for this case. The \(w=2\) case is particularly important because:

  • \(w=1\) means no masking (exact cause identification)
  • \(w=m\) means complete masking (no diagnostic information)
  • \(w=2\) is the simplest non-degenerate masking scenario

4. Asymptotic Distribution Theory

The MLE follows:

$$\sqrt{n}(\hat{\boldsymbol{\lambda}}_n - \boldsymbol{\lambda}^\star) \xrightarrow{d} \mathcal{N}(\mathbf{0}, \mathcal{I}^{-1}(\boldsymbol{\lambda}^\star))$$

with explicit Wald-type confidence intervals using the closed-form Fisher information.

Why Exponential?

The exponential distribution assumption isn’t just for tractability:

  • Constant hazard rate models systems subject to random external shocks
  • Memoryless property simplifies the likelihood structure
  • Foundation for generalization to Weibull and other distributions

More importantly, exponential is the one case where closed-form solutions exist. The insights gained here guide numerical methods for more complex distributions.

Connection to Other Work

This paper is the analytical foundation for the generalized Weibull treatment in my master’s thesis, which handles:

  • Shape and scale parameters
  • More realistic masking models
  • Model selection between candidate mechanisms

The R packages likelihood.model and likelihood.model.series.md implement these methods.

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