Inferences on Lifetime Performance Index of Weibull Products Under Step-Stress Setup

Tests based on the exponential distribution provide inaccurate assessments of the lifetime performance index (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C_{L}$</tex-math></inline-formula>) for products with nonconstant hazard rate. To address this issue, a hypothesis testing procedure is proposed for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C_{L}$</tex-math></inline-formula> under a hazard rate-based step-stress model with repeated Type II censoring, assuming a two-parameter Weibull distribution that accounts for both increasing and decreasing hazard rates. This procedure can be easily applied to assess whether the lifetime of products meets the requirements. The maximum likelihood (ML) estimator, the uniformly minimum variance unbiased estimator, and the method of moment estimator of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C_{L}$</tex-math></inline-formula> are derived and compared, highlighting the advantages of the ML estimator. For known shape parameter, hypothesis tests using the ML estimator are presented, whereas for unknown shape parameter, two bootstrap-based techniques are implemented. In addition, lower bounds for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C_{L}$</tex-math></inline-formula> are established. The robustness of the suggested process is shown by validating it on two real-life datasets and a simulated dataset. This highlights the risk of overestimating product lifetime when applying exponential distribution assumptions to data with nonconstant hazards.