Percentiles of sums of heavy-tailed random variables: beyond the single-loss approximation

A perturbative approach is used to derive approximations of arbitrary order to estimate high percentiles of sums of positive independent random variables that exhibit heavy tails. Closed-form expressions for the successive approximations are obtained both when the number of terms in the sum is deterministic and when it is random. The zeroth order approximation is the percentile of the maximum term in the sum. Higher orders in the perturbative series involve the right-truncated moments of the individual random variables that appear in the sum. These censored moments are always finite. As a result, and in contrast to previous approximations proposed in the literature, the perturbative series has the same form regardless of whether these random variables have a finite mean or not. For high percentiles, and specially for heavier tails, the quality of the estimate improves as more terms are included in the series, up to a certain order. Beyond that order the convergence of the series deteriorates. Nevertheless, the approximations obtained by truncating the perturbative series at intermediate orders are remarkably accurate for a variety of distributions in a wide range of parameters.     

Discusion

Resumen

En esta investigación se estudia la validez de una versión castellana de la escala «Environmental Concern Scale» (ECS) llevada a cabo con 716 estudiantes. Las puntuaciones obtenidas por los sujetos proambientalistas y no proambientalistas resultan significativamente diferentes. Asímismo, se estudian las relaciones entre las actitudes hacia la preocupación por el medio ambiente, la energía nuclear y la ideología política, mostrándose una relación entre ellas de diferente intensidad. Mientras la energía nuclear está fuertemente vinculada a la ideología política, no sucede en tal medida entre ésta y las actitudes conservacionistas, ya que tanto los de derechas como los de izquierdas participan de la preocupación por el medio ambiente, aunque no en la misma medida.     

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model.

We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.     

ARTIFICIAL STATES

We define artificial states as those in which political borders do not coincide with a division of nationalities desired by the people on the ground. We propose and compute for most countries in the world two measures of the degree to which borders may be artificial. One measures how borders split ethnic groups into two separate adjacent countries. The other measures the straightness of land borders, under the assumption the straight land borders are more likely to be artificial. We then show that these two measures are correlated with several measures of political and economic success.     

FAMILY TIES AND POLITICAL PARTICIPATION

We establish an inverse relationship between family ties and political participation, such that the more individuals rely on the family as a provider of services, insurance, and transfer of resources, the lower is one's civic engagement and political participation. We also show that strong family ties appear to be a substitute for generalized trust, rather than a complement to it. These three constructs—civic engagement, political participation, and trust—are part of what is known as social capital; therefore, in this paper, we contribute to the investigation of the origin and evolution of social capital. We establish these results using within‐country evidence and looking at the behavior of immigrants from various countries in 32 different destination places.     

Shortening the distance between Edgeworth and Berry–Esseen in the classical case

We obtain near optimal Berry–Esseen bounds for standardized sums of independent identically distributed random variables. This is achieved by distinguishing the lattice and the non-lattice cases, as one-term Edgeworth expansions do. The main tool is an easy inequality involving the usual second modulus of continuity, in substitution of Esseen's smoothing inequality. An illustrative example concerning the exponential distribution is also considered.