Two topics are presented in this dissertation: (1) obtaining bathtub-shaped failure rates from mixture models; and (2) the Simes inequality under dependence.;The second topic is concerned with the area of multiple testing, but uses dependence concepts important in reliability. Simes [1986] considered an improved Bonferroni test procedure based on the so-called Simes inequality. It has been proved that this inequality holds for independent multivariate distributions and a wide class of positively dependent distributions. However, as we show in this dissertation, the inequality reverses for a broad class of negatively dependent distributions. We also make some comments with regard to the Simes inequality and positive dependence.;The first topic is in the area of reliability theory. Bathtub-shaped failure rates are well-known in reliability due to their extensive applications for many electronic components, systems, products and even biological organisms. Here we derive some the conditions for obtaining bathtub-shaped failure rates distributions from mixtures, which have been utilized to model heterogeneous populations. In particular, we show that the mixtures of a family of exponential distributions and an IFR gamma distribution can yield distributions with bathtub-shaped failure rates.
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