Statistical Inference for Series Systems from Masked Failure Time Data: The Exponential Case

The analysis of masked failure data has its roots in competing risks theory. Cox [4] introduced the latent failure times model for analyzing exponentially distributed lifetimes with multiple failure types, establishing the foundational framework for modeling situations where multiple competing causes may lead to system failure. This work demonstrated that under exponential assumptions, the failed component identity and system lifetime can be independent random variables, a property that proves crucial for the analysis in this paper.

Miyakawa [12] pioneered the application of competing risks methods specifically to masked system failure data, deriving closed-form maximum likelihood estimates for a two-component series system with exponential component lifetimes. Usher and Hodgson [15] extended this approach, introducing general maximum likelihood methods for estimating component reliability from masked system life-test data and providing computational procedures for series systems. Lin et al. [11] further developed exact maximum likelihood estimation procedures for three-component systems under more general masking scenarios.

Dinse [5] developed nonparametric methods for partially-complete time and type of failure data, introducing an iterative algorithm yielding distribution-free estimates that converge to maximum likelihood solutions. This work established that useful inference is possible even when the failure cause is only partially observed.

The 1990s saw significant development of Bayesian approaches to masked data. Reiser et al. [13] introduced Bayesian inference methods for masked system lifetime data, while Guttman et al. [9] addressed dependent masking scenarios where the probability of masking depends on the true cause of failure, developing Bayesian methodology for two-component systems. These Bayesian approaches provided posterior distributions for component reliabilities but did not derive closed-form expressions for frequentist estimators or their asymptotic properties.

Flehinger et al. [6, 7] contributed to the theoretical foundations by studying survival analysis with competing risks and masked failure causes, developing parametric models that accommodate general patterns of missing failure types. Sarhan [14] provided both maximum likelihood and Bayes procedures for estimating component reliabilities in series systems with exponentially distributed lifetimes, focusing on practical implementation issues.

More recent work has expanded the scope to include interval-censored data [8], Bayesian modeling frameworks [10], and applications to specific engineering contexts. The competing risks framework continues to provide a natural setting for masked data analysis [1], connecting reliability engineering with survival analysis and biostatistics.

Despite this substantial body of work, closed-form analytical results remain limited. Specifically:

• •

  The Fisher information matrix for masked system data has not been derived in closed form for general series systems, even under exponential assumptions

• •

  Asymptotic properties of maximum likelihood estimators have not been fully characterized without numerical computation

• •

  Minimal sufficient statistics have not been identified for general masking patterns

• •

  Most existing methods rely on numerical optimization or EM algorithms without explicit characterization of estimator variance and covariance structure

This paper addresses these gaps by providing complete analytical results for the exponential case. The exponential distribution is a cornerstone of reliability theory [2], justified in several contexts: systems subject to random external shocks, systems with constant hazard rates, and early-life failure analysis where wear-out has not yet begun. The exponential assumption also serves as a limiting case for other distributions and provides a baseline for comparison with more complex models. More importantly, the exponential case admits closed-form solutions that provide direct insight into the information structure of masked failure data.

This paper provides the first complete analytical treatment of maximum likelihood estimation for series systems with masked failure data under exponential component lifetimes. Our main contributions are:

1. 1.

   Explicit Fisher information matrix: We derive a closed-form expression for the Fisher information matrix under arbitrary masking patterns, enabling direct computation of asymptotic variances without numerical differentiation or Monte Carlo simulation

2. 2.

   Sufficient statistics: The mean system lifetime and candidate set frequency vector constitute sufficient statistics, reducing the data to $1+\left(\genfrac{}{}{0pt}{}{m}{w}\right)$ real numbers

3. 3.

   Closed-form MLE for three-component systems with pairwise masking: For the important special case of $m=3$ components with candidate sets of size $w=2$, we derive an explicit closed-form solution to the likelihood equations, eliminating the need for numerical optimization. This represents the first known closed-form MLE for non-trivial masking scenarios

4. 4.

   Asymptotic distribution theory: We characterize the asymptotic sampling distribution of the MLE and provide Wald-type confidence intervals with explicit formulas for coverage

5. 5.

   Numerical validation: We provide Monte Carlo evidence confirming the theoretical asymptotic covariance matches empirical behavior for finite samples

The closed-form results for $m=3$, $w=2$ are particularly significant because this configuration represents the simplest non-degenerate masking scenario: when $w=1$ there is no masking (exact cause identification), and when $w=m$ there is complete masking (no diagnostic information). The $w=2$ case captures the essential statistical structure of masked data while admitting analytical solutions.

Section 2 introduces the probabilistic model for series systems with masked failures. Section 3 develops the likelihood and Fisher information for general parametric families. Section 4 presents the main results for exponentially distributed components, including the MLE, information matrix, and sufficient statistics. Section 5 provides detailed analysis of three-component systems. Section 6 concludes with discussion.

Consider a system composed of $m$ components. Component $j$ has a random lifetime ${\mathrm{T}}_j>0$ for $j=1,\mathrm{\dots },m$. We make the following assumptions:

Under these assumptions, the system lifetime is:

Let $F_j(t)$ and $f_j(t)$ denote the CDF and PDF of component $j$. Define the reliability function $R_j(t)=1-F_j(t)=\mathrm{P}\{{\mathrm{T}}_j>t\}$.

The system reliability function is:

The system PDF is:

When the system fails at time $t$, exactly one component caused the failure. Let $\mathrm{K}\in \{1,\mathrm{\dots },m\}$ denote the failed component. In many applications, $\mathrm{K}$ cannot be observed directly. Instead, we observe a candidate set $\mathrm{C}\subseteq \{1,\mathrm{\dots },m\}$ that contains the failed component but does not uniquely identify it.

This assumption implies that the masking quality does not depend on when the failure occurred, only on which component failed and the diagnostic capabilities of the inspection process.

We assume component $j$ has a lifetime distribution from a parametric family indexed by parameter ${\theta }_j^{\star }$. The system parameter is ${\mathrm{\Theta }}^{\star }=({\theta }_1^{\star },\mathrm{\dots },{\theta }_m^{\star })$.

Our data consists of a random sample of $n$ independent masked system failure times:

where we condition on fixed cardinality $w$ for simplicity.

Given the model assumptions, the conditional probability that component $k$ failed given system failure at time $t$ is:

Under the uniform masking model (Definition 2.3) with fixed cardinality $w$, we derive the conditional probability of observing candidate set $\mathrm{C}$ given system failure at time $t$.

This result shows that the candidate set probability is proportional to the sum of individual component failure probabilities over all components in the candidate set, normalized by the binomial coefficient representing the number of candidate sets containing each component.

The joint density of system failure time and candidate set is:

The likelihood function for sample ${𝐌}_n$ is:

The Fisher information matrix quantifies the expected information about ${\mathrm{\Theta }}^{\star }$ contained in a single observation. Under regularity conditions, the $(i,j)$-th element is:

For a sample of $n$ observations, the information is additive:

Suppose component $j$ has an exponentially distributed lifetime with failure rate ${\lambda }_j^{\star }$, denoted ${\mathrm{T}}_j\sim \mathrm{EXP}({\lambda }_j^{\star })$. The component has:

where $t>0$ and ${\lambda }_j^{\star }>0$.

The system PDF is:

An important property of exponential series systems is that $\mathrm{K}$ and $\mathrm{S}$ are independent:

The joint density is:

Under the uniform masking model (Definition 2.3), the joint density of candidate set and system failure time is:

This follows directly from Proposition 3.1 by substituting the exponential PDFs and noting that for exponential distributions, ${\prod }_{k\ne j}{R}_k(t)=\exp (-{\sum }_{k\ne j}{\lambda }_k^{\star }t)$.

The likelihood function for sample ${𝐌}_n$ with candidate sets of cardinality $w$ is:

The log-likelihood is:

The score function has $j$-th component:

where ${\mathrm{𝟙}}_{\mathrm{C}}(j)$ is the indicator function equal to 1 if $j\in \mathrm{C}$ and 0 otherwise.

In general, this requires numerical solution. However, for specific cases (notably three-component systems with $w=2$), closed-form solutions exist.

The sufficiency result shows that all information about ${𝝀}^{\star }$ in the sample is captured by: (1) the average system lifetime, and (2) the frequencies of each candidate set. We do not establish minimality: the likelihood is not a regular exponential family in $𝝀$ because the natural parameters involve $\ln ({\sum }_{j\in \mathrm{C}}{\lambda }_j)$, so a separate argument would be needed to show that no further reduction is possible.

The $(j,k)$-th element of the Fisher information matrix for exponential series systems is:

where the sum is over all candidate sets $\mathrm{C}$ of cardinality $w$.

This formula reveals that information about the pair $({\lambda }_j,{\lambda }_k)$ accrues only from candidate sets containing both components $j$ and $k$ (the indicator ${\mathrm{𝟙}}_{\mathrm{C}\times \mathrm{C}}(j,k)$ ensures this). Each such candidate set contributes inversely to its total failure rate, so components with higher failure rates contribute less information per observation. The denominator scales with the total failure rate and the number of possible candidate sets.

The following regularity conditions ensure consistency and asymptotic normality of the MLE:

1. (R1)

   The true parameter ${𝝀}^{\star }$ lies in the interior of the parameter space $\mathrm{\Theta }={(0,\mathrm{\infty })}^m$

2. (R2)

   The Fisher information matrix $ℐ({𝝀}^{\star }|w)$ is positive definite

3. (R3)

   The observations ${({\mathrm{t}}_i,{\mathrm{C}}_i)}_{i=1}^n$ are independent and identically distributed

Condition (R1) excludes boundary cases where components have zero failure rate. Condition (R2) is satisfied when each component appears in at least one candidate set, ensuring all parameters are identifiable. Condition (R3) holds by the experimental design.

This result follows from standard maximum likelihood theory [3]. The asymptotic variance-covariance matrix is the inverse of the Fisher information matrix.

An asymptotic $(1-\alpha )\times 100\%$ confidence interval for ${\lambda }_j^{\star }$ is:

where $z_{1-\alpha /2}$ is the $(1-\alpha /2)$-quantile of the standard normal distribution.

Consider observations where each candidate set has cardinality $w=2$. The three possible candidate sets are $\{1,2\}$, $\{1,3\}$, and $\{2,3\}$.

The log-likelihood is:

The score equations are:

The Fisher information matrix is:

The inverse (asymptotic variance-covariance) is obtained by symbolic matrix inversion:

This can be verified by direct multiplication: $ℐ\cdot {ℐ}^{-1}={𝐈}_3$.

The asymptotic mean squared error (trace of variance-covariance) is:

When $w=1$, each observation identifies the exact failed component. This represents the no-masking case. The MLE is:

The Fisher information matrix is diagonal:

The asymptotic variance-covariance is:

The MSE is:

which is exactly $1/3$ the MSE when $w=2$, reflecting the additional information from exact component identification.

We validate the asymptotic theory through Monte Carlo simulation studies. For each configuration, we generate $r$ independent samples of size $n$, compute the MLE for each sample, and compare the empirical covariance with the theoretical asymptotic covariance.

This work makes two key assumptions that limit its direct applicability but enable analytical tractability:

Uniform Masking Model: The assumption that non-failed components are equally likely to be included in the candidate set (Definition 2.3) is not realistic in most practical scenarios. Real diagnostic processes exhibit systematic biases:

• •

  Spatial clustering: Components in the same physical location or subsystem are often grouped in candidate sets

• •

  Accessibility bias: Easily inspected components may be overrepresented in candidate sets

• •

  Failure mode correlation: Components with similar failure signatures are more likely to be co-nominated

• •

  Cost-based selection: Expensive diagnostic tests may systematically exclude certain components

When the uniform masking assumption is violated, the likelihood in Section 3 is misspecified: the candidate-set term (7) is wrong. The resulting MLE converges to a pseudo-true parameter that solves the expected score under the incorrect masking model, which generally differs from the real parameter unless the non-uniform masking probabilities happen to be proportional across components. Intuitively, non-uniform masking up- or down-weights certain components in the log-likelihood, so the estimator absorbs masking bias instead of recovering the true failure rates. The Fisher information matrix derived here thus gives efficiency and variance only under uniform masking; with non-uniform masking, both bias and variance depend on the (unmodeled) masking mechanism.

Exponential Lifetimes: The constant-hazard assumption is appropriate for:

• •

  Random external shocks (e.g., power surges, environmental failures)

• •

  Systems without aging or wear-out (e.g., early-life failures)

• •

  Components with memoryless failure processes

The exponential assumption fails when:

• •

  Components exhibit wear-out (increasing hazard rate)

• •

  Infant mortality dominates (decreasing hazard rate)

• •

  Failure mechanisms are time-dependent

For non-exponential distributions (Weibull, gamma, lognormal), the likelihood equations generally do not admit closed-form solutions, and the Fisher information matrix must be computed numerically or via simulation.

Justification: Despite these limitations, we adopt these assumptions because they enable complete analytical solutions. The resulting formulas provide:

1. 1.

   Baseline performance bounds for assessing more complex models

2. 2.

   Insight into the fundamental information structure of masked data

3. 3.

   Computational efficiency for quick parameter estimation

4. 4.

   A reference model analogous to the role of the normal distribution in statistics

Practitioners should validate the uniform masking assumption by examining diagnostic process documentation or performing goodness-of-fit tests on candidate set patterns. When the assumption is questionable, simulation studies can assess robustness or alternative models (e.g., component-specific masking probabilities) should be considered.

Several extensions would broaden the applicability of this framework:

1. 1.

   Non-uniform masking models: Relax the uniform masking assumption to allow component-specific inclusion probabilities ${\alpha }_j$ or conditional dependence structures. For example:

   • •

     Accessibility-weighted masking: $\mathrm{P}\{j\in \mathrm{C}|\mathrm{K}=k,j\ne k\}={\alpha }_j$ depends on component accessibility

   • •

     Cluster-based masking: Components within the same subsystem have correlated inclusion probabilities

   • •

     Parametric masking models with estimable diagnostic parameters

   Such models sacrifice analytical tractability but more accurately reflect real diagnostic processes. The uniform masking results provide baseline comparisons and limiting cases.

2. 2.

   Variable masking cardinality: Allow $w$ to vary across observations, modeling scenarios where diagnostic quality varies by failure instance or by component

3. 3.

   Non-exponential lifetime distributions: Extend to Weibull (aging/wear-out), gamma (multi-stage failures), or lognormal (time-dependent hazards) distributions. Numerical methods will be required, but asymptotic theory remains applicable.

4. 4.

   Covariate effects: Incorporate covariate information (operating conditions, environmental factors) via proportional hazards models or accelerated failure time frameworks

5. 5.

   Bayesian approaches: When prior information about component reliabilities is available, derive posterior distributions for $𝝀$ and posterior predictive distributions for future failures

6. 6.

   Optimal diagnostic design: Given costs of diagnostic tests and benefits of precise failure identification, design inspection strategies that optimize the information-cost tradeoff

The uniform masking, exponential lifetime framework provides a foundation for understanding masked system data. While restrictive, the analytical tractability enables clear insight into the information structure that extends conceptually to more complex settings. The closed-form results serve as benchmarks for evaluating numerical methods and assessing the efficiency loss from model misspecification.