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Robust Rao-type tests for step-stress accelerated life-tests under interval-monitoring and Weibull lifetime distributions

Narayanaswamy Balakrishnan, María Jaenada, Leandro Pardo

Abstract

Many products in engineering are highly reliable with large mean lifetimes to failure. Performing lifetests under normal operations conditions would thus require long experimentation times and high experimentation costs. Alternatively, accelerated lifetests shorten the experimentation time by running the tests at higher than normal stress conditions, thus inducing more failures. Additionally, a log-linear regression model can be used to relate the lifetime distribution of the product to the level of stress it experiences. After estimating the parameters of this relationship, results can be extrapolated to normal operating conditions. On the other hand, censored data is common in reliability analysis. Interval-censored data arise when continuous inspection is difficult or infeasible due to technical or budgetary constraints. In this paper, we develop robust restricted estimators based on the density power divergence for step-stress accelerated life-tests under Weibull distributions with interval-censored data. We present theoretical asymptotic properties of the estimators and develop robust Rao-type test statistics based on the proposed robust estimators for testing composite null hypothesis on the model parameters.

Robust Rao-type tests for step-stress accelerated life-tests under interval-monitoring and Weibull lifetime distributions

Abstract

Many products in engineering are highly reliable with large mean lifetimes to failure. Performing lifetests under normal operations conditions would thus require long experimentation times and high experimentation costs. Alternatively, accelerated lifetests shorten the experimentation time by running the tests at higher than normal stress conditions, thus inducing more failures. Additionally, a log-linear regression model can be used to relate the lifetime distribution of the product to the level of stress it experiences. After estimating the parameters of this relationship, results can be extrapolated to normal operating conditions. On the other hand, censored data is common in reliability analysis. Interval-censored data arise when continuous inspection is difficult or infeasible due to technical or budgetary constraints. In this paper, we develop robust restricted estimators based on the density power divergence for step-stress accelerated life-tests under Weibull distributions with interval-censored data. We present theoretical asymptotic properties of the estimators and develop robust Rao-type test statistics based on the proposed robust estimators for testing composite null hypothesis on the model parameters.
Paper Structure (7 sections, 3 theorems, 57 equations, 4 figures)

This paper contains 7 sections, 3 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Weibull lifetime distribution
  3. Minimum restricted density power divergence estimator
  4. Robustness analysis
  5. Robust Rao-type test statistics
  6. Reliability analysis for solar lighting devices
  7. Concluding remarks

Key Result

Theorem 1

The RMDPDE of the interval-censored step-stress ALT model under Weibull lifetime distributions must satisfy the following system of $3+ r$ equations where $\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta})}$ denotes a $(L+1)\times(L+1)$ diagonal matrix with diagonal entries $\pi_j(\boldsymbol{\theta}),$$j=1,...,L+1,$ and $\boldsymbol{W}(\boldsymbol{\theta})$ is a $(L+1) \times 3$ matrix with

Figures (4)

  • Figure 1: Mean squared error (MSE) of the model estimates (left) and mean time to failure (right) for different values of the tuning parameter $\beta$ under increasing contamination rates
  • Figure 2: Mean squared error (MSE) of the RMDPDE for different values of the tuning parameter $\beta$ under increasing contamination rates
  • Figure 3: Empirical level of the Rao-type test statistics based on the RMDPDEs for different values of the tuning parameter $\beta$ under increasing contamination rates
  • Figure 4: Empirical power of the Rao-type test statistics based on the RMDPDEs for different values of the tuning parameter $\beta$ under increasing contamination rates

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 4