Tampered random variable modeling for multiple step-stress life test

Abstract

In this paper, we introduce the tampered random variable (TRV) modeling in multiple step-stress life testing experiments. Here ${\tau }_1<{\tau }_2<\dots <{\tau }_{k-1}$ be $(k-1)$ prespecified time points and $s_1,s_2,\dots ,s_k$ be k prefixed stress levels with s_i being the stress level in force during the time interval $[{\tau }_{i-1},{\tau }_i)$ for $i=1,\dots ,k$ with ${\tau }_0=0$ and ${\tau }_k=\infty .$ We define the tampered random variable $T_{\mathit{TRV}}^{(k)}$ in multiple step-stress scenario and calculate the PDF, CDF, and Hazard rate for the proposed tampered variable $T_{\mathit{TRV}}^{(k)}.$ We derive a general expression for the expectation of $T_{\mathit{TRV}}^{(k)}$ under different number k of stress levels and also obtain some results on stochastic ordering for different k. All these results are obtained under arbitrary baseline (under normal stress condition with stress level s₁) life distribution. In particular, we consider exponential distribution with mean θ and Weibull distribution with scale parameter λ and shape parameter α for specific expressions. We also prove some results on equivalence of the TRV modeling with the two other existing models for step-stress life testing, namely, cumulative exposure and tampered failure rate. Finally, we consider some variations of the modeling approach for $T_{\mathit{TRV}}^{(k)}$ to include incorporation of the stress levels, discrete life time, bivariate or multivariate life times.

Keywords:

• Life testing experiments
• tampering coefficients
• tampering times
• stress conditions
• baseline lifetime
• stochastic ordering

Acknowledgement

The authors are thankful to the anonymous reviewer for many helpful comments which have improved the writing of this paper.