Principios de prima y medidas de desigualdad basados en mixturas de estadísticos ordenados

Abstract

In applied probability and, more specifically, in decision-making processes, a probability distribution is often transformed before taking expectations to reflect the attitude or perception of decision-makers. In actuarial theory, for example, risk-adjusted premiums are often defined via transformations of the risk distribution to incorporate the degree of risk-aversion of the insurer. Similarly, in economics, inequality and welfare measures incorporate the magnitude of inequality-aversion via transformations of the income distribution. Following this approach, we propose in this thesis a method to adjust probability distributions based on mixtures of order statistics.

In Chapter 2 we give a general method for constructing risk-adjusted distributions. We claim that an appropriate risk-adjusted distribution, besides satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted distributions, we introduce a family of premium principles that gradually incorporate the degree of risk-aversion of the insurer in the risk loading. Members of this family are particular distortion premium principles that can be represented as mixtures of tail value-at-risks with beta mixing distributions, where the weights in the mixture reflect the attitude toward risk of the insurer. We make a systematic study of this family of premium principles.

In Chapter 3 we show that every premium principle in this family coincides with the expected average of the n-i largest claims, with 0 ≤ i ≤ n-1, of a set of n≥ 2 independent and identically distributed claims. From this result, we interpret the tail value-at-risk in terms of the largest claims of a portfolio of independent claims. As an application, we provide sufficient conditions for stochastic comparisons of premiums in the context of large claims reinsurance.

In Chapter 4, a general method for constructing risk-adjusted distributions that extends the one described previously is presented. The method is based on modifying the underlying risk distribution in such a way that the risk-adjusted expected value (or premium principle) is greater than the expected value of some conveniently chosen function of claims, which defines the insurer's perception of the risk. This method generalizes the family of premium principles considered in Chapter 2, which was restricted to the case where the functions of claims are order statistics. We study in this chapter the cases where the functions of claims are record claims and k-record claims, producing special families of distortion premium principles.

In Chapter 5 we provide sufficient conditions on two income random variables for comparing, in terms of the increasing concave order, some random variables describing social welfare based on linear combinations of their order statistics.

Finally, in Chapter 6 we present the conclusions.