Testing ageing notions through percentiles of the residual life

In applications, it is common to classify life distributions according to different ageing properties based on different reliability measures. Those based on the failure rate function or the mean residual life are very popular in the literature. However, the increasing or decreasing failure rate ageing notion is very restrictive and, on the other hand, the mean residual life function has some drawbacks which make convenient to prevent its use: it may not exist and, even when it does, sometimes the empirical mean residual life function cannot be calculated or a single long-term survivor can have a marked effect upon it which will tend to be unstable due to its strong dependence on very long durations. The median, or other percentiles, of the residual life of a random variable, are, in general, useful alternatives to the mean residual life. In this paper, we propose a test statistic to check the ageing notion decreasing percentile residual life. This estimator is based on the difference of the empirical percentile residual life function and the monotone estimator. We study the asymptotic and the bootstrap distributions of the proposed statistic. A Monte Carlo simulation study has been carried out to see the performance of the statistic in finite sample situations. Finally, we apply our procedure to a real data set.