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Likelihood-Based Regression for Weibull Accelerated Life Testing Model Under Censored Data

Rahul Konar, Ramnivas Jat, Neeraj Joshi, Raghu Nandan Sengupta

TL;DR

The paper develops a likelihood-based two-step framework for Weibull ALT models with stress-dependent shape and scale under PHC and APHC censoring. It proves consistency and asymptotic normality of the ML estimators for the Weibull parameters and then uses a regression stage (with Murphy–Topel corrections) to estimate stress coefficients linking $\alpha$ and $\lambda$ to temperature $T$ and voltage $V$. The approach is validated through extensive simulations showing accurate, robust parameter recovery and precise stress-coefficient estimates, and a data illustration demonstrates practical applicability and interpretability of the stress–life relationships. Overall, the work offers a flexible, theoretically grounded methodology for modeling stress-dependent reliability under complex censoring in ALT studies.

Abstract

In this paper, we investigate accelerated life testing (ALT) models based on the Weibull distribution with stress-dependent shape and scale parameters. Temperature and voltage are treated as stress variables influencing the lifetime distribution. Data are assumed to be collected under Progressive Hybrid Censoring (PHC) and Adaptive Progressive Hybrid Censoring (APHC). A two-step estimation framework is developed. First, the Weibull parameters are estimated via maximum likelihood, and the consistency and asymptotic normality of the estimators are established under both censoring schemes. Second, the resulting parameter estimates are linked to the stress variables through a regression model to quantify the stress-lifetime relationship. Extensive simulations are conducted to examine finite-sample performance under a range of parameter settings, and a data illustration is also presented to showcase practical relevance. The proposed framework provides a flexible approach for modeling stress-dependent reliability behavior in ALT studies under complex censoring schemes.

Likelihood-Based Regression for Weibull Accelerated Life Testing Model Under Censored Data

TL;DR

The paper develops a likelihood-based two-step framework for Weibull ALT models with stress-dependent shape and scale under PHC and APHC censoring. It proves consistency and asymptotic normality of the ML estimators for the Weibull parameters and then uses a regression stage (with Murphy–Topel corrections) to estimate stress coefficients linking and to temperature and voltage . The approach is validated through extensive simulations showing accurate, robust parameter recovery and precise stress-coefficient estimates, and a data illustration demonstrates practical applicability and interpretability of the stress–life relationships. Overall, the work offers a flexible, theoretically grounded methodology for modeling stress-dependent reliability under complex censoring in ALT studies.

Abstract

In this paper, we investigate accelerated life testing (ALT) models based on the Weibull distribution with stress-dependent shape and scale parameters. Temperature and voltage are treated as stress variables influencing the lifetime distribution. Data are assumed to be collected under Progressive Hybrid Censoring (PHC) and Adaptive Progressive Hybrid Censoring (APHC). A two-step estimation framework is developed. First, the Weibull parameters are estimated via maximum likelihood, and the consistency and asymptotic normality of the estimators are established under both censoring schemes. Second, the resulting parameter estimates are linked to the stress variables through a regression model to quantify the stress-lifetime relationship. Extensive simulations are conducted to examine finite-sample performance under a range of parameter settings, and a data illustration is also presented to showcase practical relevance. The proposed framework provides a flexible approach for modeling stress-dependent reliability behavior in ALT studies under complex censoring schemes.
Paper Structure (23 sections, 6 theorems, 77 equations, 12 figures, 17 tables)

This paper contains 23 sections, 6 theorems, 77 equations, 12 figures, 17 tables.

Table of Contents

  1. Introduction and Background
  2. The Weibull ALT Model
  3. Statistical Framework of PHC
  4. Statistical Framework of APHC
  5. ML Estimation of the Model Parameters
  6. ML Estimation Under PHC
  7. Case 1: When all $m$ failures observed before time $T$
  8. Case 2: When exactly $j$ out of $m$ failures occur before time $T$
  9. ML Estimation Under APHC
  10. Case 1: When $m$ failures are observed before time $T$
  11. Case 2: When exactly $j$ failures occur before time $T$
  12. Consistency of $\hat{\alpha}_{ML}$ Under PHC and APHC Case I
  13. Asymptotic Normality of $\hat{\alpha}_{ML}$ Under PHC and APHC Case I
  14. Consistency of $\hat{\alpha}_{ML}$ Under PHC Case II
  15. Consistency of $\hat{\alpha}_{ML}$ Under APHC Case II
  16. ...and 8 more sections

Key Result

Theorem 1

Let $X_1,\dots,X_n$ be i.i.d. random variables following a two-parameter Weibull distribution with shape parameter $\alpha_0>0$ and scale parameter $\lambda>0$. Consider a pre-fixed PHC scheme $(R_1,\dots,R_m)$ satisfying $R_1+\cdots+R_m+m=n$. Let $x_{(1)}<x_{(2)}<\cdots<x_{(m)}$ denote the ordered Assume that $m=m_n\to\infty$ as $n\to\infty$, and that the censoring weights $w_i=(1+R_i)/\sum_{j=1

Figures (12)

  • Figure 1: Dataset 3: PHC Shape
  • Figure 2: Dataset 9: PHC Shape
  • Figure 3: Dataset 15: PHC Shape
  • Figure 4: Dataset 3: PHC Scale
  • Figure 5: Dataset 9: PHC Scale
  • ...and 7 more figures

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof : Proof
  • Remark 2
  • Lemma 1: Uniqueness and Monotonicity
  • proof
  • Lemma 2: Uniform convergence
  • proof
  • ...and 4 more