Conditional tail expectation of randomly weighted sums with heavy-tailed distributions

2015 ◽  
Vol 105 ◽  
pp. 20-28 ◽  
Author(s):  
Yang Yang ◽  
Eglė Ignatavičiūtė ◽  
Jonas Šiaulys
Keyword(s):  
Weighted Sums ◽  
Randomly Weighted Sums ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Conditional Tail Expectation

A note on the max-sum equivalence of randomly weighted sums of heavy-tailed random variables

2013 ◽  
Vol 18 (4) ◽  
pp. 519-525 ◽  
Author(s):  
Yang Yang ◽  
Kaiyong Wang ◽  
Remigijus Leipus ◽  
Jonas Šiaulys
Keyword(s):  
Random Variables ◽  
Tail Probability ◽  
Weighted Sums ◽  
Asymptotic Formulas ◽  
Dependence Structure ◽  
Insurance Risk ◽  
Risk Models ◽  
Randomly Weighted Sums ◽  
Random Weights ◽  
Heavy Tailed

This paper investigates the asymptotic behavior for the tail probability of the randomly weighted sums ∑k=1nθkXk and their maximum, where the random variables Xk and the random weights θk follow a certain dependence structure proposed by Asimit and Badescu [1] and Li et al. [2]. The obtained results can be used to obtain asymptotic formulas for ruin probability in the insurance risk models with discounted factors.

scholarly journals Improved Estimator of the Conditional Tail Expectation in the case of heavy-tailed losses

2020 ◽  
Vol 8 (1) ◽  
pp. 98-109
Author(s):  
Mohamed Laidi ◽  
Abdelaziz Rassoul ◽  
Hamid Ould Rouis
Keyword(s):  
Asymptotic Normality ◽  
Simulation Study ◽  
Mean Squared Error ◽  
Extreme Value ◽  
Loss Distribution ◽  
Squared Error ◽  
Heavy Tailed Distributions ◽  
High Quantile ◽  
Heavy Tailed ◽  
Conditional Tail Expectation

In this paper, we investigate the extreme-value methodology, to propose an improved estimator of the conditional tail expectation (CTE) for a loss distribution with a finite mean but infinite variance.The present work introduces a new estimator of the CTE based on the bias-reduced estimators of high quantile for heavy-tailed distributions. The asymptotic normality of the proposed estimator is established and checked, in a simulation study. Moreover, we compare, in terms of bias and mean squared error, our estimator with the known old estimator.

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

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