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Robust Estimation in Step-Stress Experiments under Weibull Lifetime Distributions

María Jaenada, Juan Millán, Leandro Pardo

Abstract

Many modern products are highly reliable, often exhibiting long lifetimes. As a result, conducting experiments under normal operating conditions can be prohibitively time-consuming to collect sufficient failure data for robust statistical inference. Accelerated life tests (ALTs) offer a practical solution by inducing earlier failures, thereby reducing the required testing time. In step-stress experiments, a stress factor that accelerates product degradation is identified and systematically increased at predetermined time points, while remaining constant between intervals. Failure data collected under these elevated stress levels is analyzed, and the results are then extrapolated to normal operating conditions. Traditional estimation methods for such data, such as the maximum likelihood estimator (MLE), are highly efficient under ideal conditions but can be severely affected by outlying or contaminated observations. To address this, we propose the use of Minimum Density Power Divergence Estimators (MDPDEs) as a robust alternative, offering a balanced trade-off between efficiency and resistance to contamination. The MDPDE framework is extended to mixed distributions and its theoretical properties, including the asymptotic distribution of the model parameters, are derived assuming Weibull lifetimes. The effectiveness of the proposed approach is illustrated through extensive simulation studies, and its practical applicability is further demonstrated using real-world data.

Robust Estimation in Step-Stress Experiments under Weibull Lifetime Distributions

Abstract

Many modern products are highly reliable, often exhibiting long lifetimes. As a result, conducting experiments under normal operating conditions can be prohibitively time-consuming to collect sufficient failure data for robust statistical inference. Accelerated life tests (ALTs) offer a practical solution by inducing earlier failures, thereby reducing the required testing time. In step-stress experiments, a stress factor that accelerates product degradation is identified and systematically increased at predetermined time points, while remaining constant between intervals. Failure data collected under these elevated stress levels is analyzed, and the results are then extrapolated to normal operating conditions. Traditional estimation methods for such data, such as the maximum likelihood estimator (MLE), are highly efficient under ideal conditions but can be severely affected by outlying or contaminated observations. To address this, we propose the use of Minimum Density Power Divergence Estimators (MDPDEs) as a robust alternative, offering a balanced trade-off between efficiency and resistance to contamination. The MDPDE framework is extended to mixed distributions and its theoretical properties, including the asymptotic distribution of the model parameters, are derived assuming Weibull lifetimes. The effectiveness of the proposed approach is illustrated through extensive simulation studies, and its practical applicability is further demonstrated using real-world data.

Paper Structure

This paper contains 24 sections, 13 theorems, 185 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Step-Stress Model under Weibull Distributions
  3. The minimum density power divergence estimator
  4. Asymptotic distribution of the minimum density power divergence estimator
  5. Lifetime Characteristics Estimation
  6. Simulation Study
  7. Performance of the MPDE
  8. Real-Data Example
  9. Conclusions
  10. Appendix
  11. Proof Theorem 3
  12. Proof Lemma 1
  13. Proof Proposition 2
  14. Proof Proposition 4
  15. Proof Proposition 5
  16. ...and 9 more sections

Key Result

Lemma 1

Under Weibull lifetimes, we have Where $\gamma(s,a)$ is the incomplete Gamma function:

Figures (7)

  • Figure 1: Step-stress test design with Type-I censoring.
  • Figure 2: RMSE of the MDPDEs under increasing contamination proportion for the estimation of a0, a1, eta
  • Figure 3: RMSE of $\lambda_i$ when the contaminated parameter is $a_1$.
  • Figure 4: RMSE for the MTTF (top), the Reability at t=2 (center) and the median (bottom) when the contaminated parameter is $a_1$.
  • Figure 5: Ellipse of the estimations for $a_0$ and $a_1$ for the 95% under contamination in the parameter $a_1$.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Definition 3
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 20 more