Series expansions for convolutions of Pareto distributions

IF 1.3 Q2 STATISTICS & PROBABILITY

Statistics & Risk Modeling Pub Date : 2015-04-01 DOI:10.1515/strm-2014-1168

Q. Nguyen, C. Robert

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引用次数: 3

Abstract

Abstract Asymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.

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帕累托分布卷积的级数展开

摘要:本文主要推导了具有正则尾变化的随机变量和的尾的渐近展开式，给出了同分布随机变量和具有相同尾指数的随机变量的尾的渐近展开式。此外，高阶项通常是在分布的一个矩存在的条件下给出的。本文给出了尾指数为非整数的Pareto分布卷积的无穷级数展开式。Pareto随机变量可能有不同的尾指数和不同的尺度参数。在同分布随机变量的情况下，我们很自然地找到了在前面的渐近展开中给出的第一项的常数，但是我们现在能够给出下一项的附加项。由于我们的级数展开式不是渐近的，它也可以用于计算总和分布的分位数值以及其他风险度量，如风险尾部值。为至少五个帕累托随机变量的和提供了值的示例，并与通过以前的渐近展开或通过模拟确定的值进行了比较。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Statistics & Risk Modeling STATISTICS & PROBABILITY-

CiteScore

1.80

自引率

6.70%

发文量

期刊介绍： Statistics & Risk Modeling (STRM) aims at covering modern methods of statistics and probabilistic modeling, and their applications to risk management in finance, insurance and related areas. The journal also welcomes articles related to nonparametric statistical methods and stochastic processes. Papers on innovative applications of statistical modeling and inference in risk management are also encouraged. Topics Statistical analysis for models in finance and insurance Credit-, market- and operational risk models Models for systemic risk Risk management Nonparametric statistical inference Statistical analysis of stochastic processes Stochastics in finance and insurance Decision making under uncertainty.