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. The accuracy of the approximations is illustrated for a variety of distributions and a wide range of parameters. The quality of the estimate improves as more terms are included in the perturbative series, specially for higher percentiles and heavier tails.
Besides the heuristic derivation given in @cite_38 and the perturbative expansion proposed in this work, higher order corrections to the single-loss approximation can be derived in at least three different ways: Using the second order asymptotic approximations introduced in @cite_19 @cite_2 @cite_27 @cite_25 , from the asymptotic expansion analyzed in @cite_31 @cite_40 @cite_35 or from asymptotic approximations based on evaluations of @math at different arguments @cite_32 .
@cite_38 @cite_35 @cite_32 @cite_19 @cite_27 @cite_40 @cite_2 @cite_31 @cite_25
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Making the assumption that the distribution of operational-loss severity has finite mean, Klaus Bocker and Jacob Sprittulla suggest a refined version of the analytical operational VAR theorem derived in Bocker and Kluppelberg (2005), which significantly reduces the approximation error to operational VAR. Let X i (i=1,2, …) be a sequence of subexponential positive independent and identically distributed random variables. In this paper, we offer two alternative approaches to obtain higher-order expansions of the tail of and subsequently for ruin probabilities in renewal risk models with claim sizes X i . In particular, these emphasize the importance of the term for the accuracy of the resulting asymptotic expansion of . Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto type, Weibull and Lognormal distribution for X i are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given. Let G = [Sigma][infinity]n=0pnF*n denote the probability measure subordinate to F with subordinator Pn . We investigate the asymptotic behaviour of (1 - G(x))-([Sigma] npn)(1 - F(x)) as x --> [infinity] if 1 - F is regularly varying with index [varrho], 0 We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions. In this paper we construct asymptotic expansions for the tail area of a compound sum, when the distribution of the summands belongs to a class of rapidly varying subexponential distributions. To be more precise, let X?, i > 1, be a sequence of independent random variables, all having the same distribution, F. For any positive integer n the distribution of the partial sums Sn = X + + Xn is the n-fold convolution F*n. We set So = 0 and therefore F*? is defined as the distribution of the point mass at the origin. Let N be a nonnegative integer-valued random variable, independent of the X s. We consider the distribution G of the compound sum Sn, that is E F*N, and we seek an asymptotic expansion for its tail area G = 1 ? G. First-order asymptotic results for G have been obtained by (1979), Cline (1987), and Embrechts (1985). A second-order formula may be found in Griibel (1987) and Omey and Willekens (1987). Compound sums or subordinated distributions arise as distributions of interest in several stochastic models. In insurance risk theory, it models the total claim amount. For a discussion of issues related to random sums and insurance risk, we refer the reader to Embrechts et In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails. @PARASPLIT Higher-order asymptotics are also obtained when considering a semiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.