Robust Life-Time Estimation based on the Density Power Divergence for Highly Reliable Products

Abstract

Handling censored data in reliability and survival analysis is a key concern in practice. Interval-censored data emerges in experiments where failure times are only known to fall within a specific interval rather than being observed precisely. Additionally, many modern devices are extremely reliable, with long lifetimes to failure, necessitating extended experimental durations for inference under normal operating conditions. Instead, accelerated life tests (ALTs) are used to reduce the lifetime of devices by increasing one or more stress factors, which induce failure. After suitable inference, results can be extrapolated to working conditions. Classical estimation methods relying on the likelihood function of the lifetime distribution can be significantly influenced by data contamination. As an alternative, robust estimators based on distance measures are developed. In this talk, we develop robust inferential methods based on the density power divergence for inferring the lifetime distribution of highly reliable products under interval censoring.

• McMaster University, Hamilton, Canada, [bala@mcmaster.ca]

• Complutense University of Madrid [mjaenada@ucm.es, lpardo@ucm.es]

Keywords: Accelerated Life-Tests – Reliability – Robustness

Extended Abstract

Inferential methods in reliability and survival analysis investigate the lifetime distribution of a product/individual often from a parametric approach. Some products are highly reliable with large lifetimes under normal operating conditions, which makes experimentation quite difficult as it would result in high cost and impractically long time. In such cases, experimenter may adopt accelerated life-testing, wherein the experimental units are subjected to higher stress levels than normal operating conditions. Moreover, censoring times may appear in many experimentation procedures. For example, due to the nature of the product or due to experimental or budget constraints, the devices under test may not be subject to continuous monitoring, but may be subject to periodic inspections. That is, when only the failure counts can be collected at certain pre-fixed time points during the test, resulting in interval-censored data. Accelerated life tests (ALTs) are used to shorten the mean lifetime of products by increasing the stress level they are subjected to. This way, sufficient data for inference can be recorded in a shorter experimental time, which can then be suitably modelled and analyzed, and the obtained results could later be extrapolated to the case of normal operating conditions. The ALT model must then relate the mean lifetime of a product to the stress factor at which the product is subjected to, frequently through a log-linear relationship. Moreover, ALTs allow one to asses the effect of stress factors, such as pressure, voltage or temperature, on the lifetimes of experimental units. ALT is usually performed using constant stress, step stress, or linearly increasing stress levels. The step-stress ALT model, in particular, increases the stress level to all surviving units at certain pre-fixed times for stress change. Then, unlike the constant stress experiment, the step-stress experiment reduces the test time by promoting quicker failures as well as a larger number of failures. Despite all the advantages of step-stress tests resulting in by quicker failures in a short test time, step-stress ALT requires a model relating the lifetime distribution at one stress level to the lifetime distribution at the preceding stress levels. One such model is the cumulative exposure model (CE), which assumes that the residual lives of the experimental units depend only on the cumulative exposure the units have experienced thus far, with no memory of how this exposure was accumulated. That is, it assumes that after holding a group of devices at a particular stress, surviving units will fail according to the distribution at the current stress, but starting with the previously accumulated failure probability that has been be accumulated under various stress levels. The CE is the most prominent and commonly used model in reliability analysis for step-stress experiments, as it is plausible in many situations that, while a product is operating, there is some damage-accumulation in the product that would impact its reliability. Particularly, the CE assumes that the product reliability depends on its wear, regardless of how it was produced or under what stress, and the level of stress to which the product had been subjected to at that time. Exponential, Weibull, gamma and log-normal distributions are some of the most common parametric families used for modelling lifetime in reliability engineering analyses. Step-stress ALTs assume that the scale parameter of any of these families is related to the stress level, but the shape parameter is common for any stress level, an assumption that simplifies the application of CE. Classical estimators and tests are based on the maximum likelihood criterion, resulting in the maximum likelihood estimator (MLE) which is asymptotically efficient but lacks robustness. That is, the MLE is a good estimator, in the absence of contamination, possessing a small variance compared to other estimators. However, it is quite sensitive to outlying observations resulting from measurement or experimental errors, and even small contamination in the data could significantly influence the performance of the MLE. In this regard, divergence-based estimation methods have proven to provide a gainful trade-off between efficiency and robustness in many other statistical reliability models under censorship. This work develop robust inferential techniques for ALT data under interval-censoring.