Robust statistical inference for accelerated life-tests with one-shot devices under log-logistic distributions

TL;DR

This work develops robust statistical inference for accelerated life tests of one-shot devices with lifetimes following a log--logistic distribution under multiple stress factors. It introduces weighted minimum density power divergence estimators with tuning parameter $\gamma$ to obtain robust parameter estimates for the stress life model $\alpha_i$ and $\beta_i$ linked via $α_i = e^{\sum_j a_j x_{ij}}$ and $β_i = e^{\sum_j b_j x_{ij}}$, and derives their asymptotic distribution $\sqrt{K}(\hat{\boldsymbol{\theta}}_{\gamma} - \boldsymbol{\theta}_0) \to N(0, \boldsymbol{J}_{\gamma}^{-1}(\boldsymbol{\theta}_0) \boldsymbol{K}_{\gamma}(\boldsymbol{\theta}_0) \boldsymbol{J}_{\gamma}^{-1}(\boldsymbol{\theta}_0))$. The paper also develops robustness diagnostics via the influence function and introduces robust Wald-type and Rao-type tests based on the WMDPDE, with explicit matrices $\boldsymbol{J}_{\gamma}$, $\boldsymbol{K}_{\gamma}$ and their restricted counterparts, both converging to $\chi^2_r$ under $H_0$. Through Monte Carlo simulations and a real data application, it demonstrates that moderate values of $\gamma$ yield substantial robustness to contamination with only modest efficiency loss, validating the method for practical ALT planning of highly reliable one-shot devices.

Abstract

A one-shot device is a unit that operates only once, after which it is either destroyed or needs to be rebuilt. For this type of device, the operational status can only be assessed at a specific inspection time, determining whether failure occurred before or after it. Consequently, lifetimes are subject to left- or right-censoring. One-shot devices are usually highly reliables. To analyze the reliability of such products, an accelerated life test (ALT) plan is typically employed by subjecting the devices to increased levels of stress factors, thus allowing life characteristics observed under high-stress conditions to be extrapolated to normal operating conditions. By accelerating the degradation process, ALT significantly reduces both the time required for testing and the associated experimental costs. Recently, robust inferential methods have gained considerable interest in statistical analysis. Among them, weighted minimum density power divergence estimators (WMDPDEs) are widely recognized for their robust statistical properties with small loss of efficiency. In this work, robust WMDPDE and associated statistical tests are developed under a log-logistic lifetime distribution with multiple stresses. Explicit expressions for the estimating equations and asymptotic distribution of the estimators are obtained. Further, a Monte Carlo simulation study is presented to evaluate the performance of the WMDPDE in practical applications.

Figures (5)

• Figure 1: Log-Logistic Distribution Functions with Fixed Scale Parameter $\alpha = 2$ and Varying Shape Parameter
• Figure 2: Root mean square errors of parameter estimates under contaminated data for different $\gamma$ values
• Figure 3: RMSE (bottom left panel), Empirical Level (top left) and Empirical Power (top right) under pure data for different numbers of devices (K)
• Figure 4: Wald-type tests empirical levels and empirical powers under contaminated data for different $\gamma$ values
• Figure 5: Estimated mean lifetime until failure for different values of $\gamma$ under normal temperature conditions ($x=\frac{1}{298}$)

Theorems & Definitions (9)

• Proposition 2
• Corollary 4
• Definition 8
• Example 9
• Definition 10
• Corollary 12
• Proposition 13
• Definition 14
• Remark 15