Condiciones suficientes para criterios de comparación estocástica

Resum

One of the main objectives of statistics is the comparison of random quantities. These comparisons are mainly based on measures associated to these random quantities, like the means, medians or variances. In some situations, the comparisons based only on two single measures are not very informative. The necessity of providing more elaborate comparisons of two random quantities has motivated the development of the theory of the stochastic orders. This theory is composed of different criteria which compare different characteristics measured by several functions of interest in reliability, risks and economy. Such functions are defined in terms of certain incomplete integrals of the survival or quantile functions. Unfortunately, these integrals can not be given in an explicit way in most cases, therefore we can not check directly these orders by their definitions. Even though we can verify some stronger orders or conditions, there are lots of cases which are not covered by any tool in the literature. Due to this fact, one of the main direction of research on this topic is the exploration of sufficient conditions, easy to verify, for these orders when the strongest ones do not hold. For instance, the well-known increasing convex order holds when the incomplete integrals of the survival functions are ordered. However, there are loads of situations where this integrals do not have analytical expression, which makes difficult to verify the order. In this case, we can check the stochastic order, since it is widely known that it is the strongest order to compare location and holds when the survival functions are ordered, which is a simple condition to verify, as long as the survival functions have an explicit expression; but also if this does not occurs, there exists a sufficient condition in terms of the density functions. However, there are lots of situations where the random variables are not ordered in the stochastic order. Luckily, to check the increasing convex order in this cases, there exist the renowned Karlin-Novikov conditions established in terms of the crossing among the survival, quantile and density functions, therefore these conditions can be always verified. Our main purpose in this thesis is to continue this line of research for some of the main orders in the literature: the mean residual life, the total time on test transform, the excess wealth and the expected proportional shortfall order. Down to the last detail, we consider the situation where the strongest orders are not verified (the hazard rate, the stochastic, the dispersive and the star shaped orders, respectively), which are mainly defined in terms of the monotonicity of the ratio (or the difference) of the survival or the quantile functions. We seek an intermediate condition between the strongest order and the corresponding weaker order. Principally, we consider the situations where they have a relative extreme, a property less restrictive than being monotone, to establish several sets of sufficient conditions. Furthermore, a relevant goal of this thesis is to apply the provided results to order several well-known parametric families, which have particular interest to fit data in reliability, risks and economy. The stochastic comparison of ordered data is another of the most important areas of research in this topic and we also provide some results in this direction. Finally, we work on the joint stochastic orders, which take into account the dependence structure among the random variables. This is of interest, for example, if we are interested in comparing two units which are aging in the same environment and both depend on this environment. We also provide some contributions to this area.