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Let F be a bathtub-shaped (BT) hazard rate distribution function. It has been shown that the hazard rate function of the order statistics may be BT, increasing, etc. Then, we have carried out a graphical study for some useful lifetime models.Moreover, we are interested to compare the time that maximizes the mean residual life (MRL) function of F with the one related to a general weighted model in terms of their locations. Also, the times maximizing the conditional reliability proposed by Mi [13] of F have been compared with the corresponding times of a general weighted model. As special cases, we consider order statistics and the proportional hazard rate model.

The Lindley distribution may serve as a useful reliability model. Applications of this distribution are presented in statistical literature. In this article, goodness of fit tests for the Lindley distribution based on the empirical distribution function (EDF) are considered. In order to compute the test statistics, we use the maximum likelihood estimate (MLE) suggested by Ghitany et al. (2008), which is simple explicit estimator. Critical points of the proposed test statistics are obtained by Monte Carlo simulation. Power comparisons of the considered tests are carried out via simulations. Finally, two illustrative examples are presented and analyzed.