Asymptotic results for heavy-tailed distributions using defective renewal equations

2009 ◽  
Vol 79 (6) ◽  
pp. 774-779 ◽  
Author(s):  
Georgios Psarrakos
Keyword(s):  
Asymptotic Results ◽  
Renewal Equations ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Defective Renewal Equations

scholarly journals Yule–Walker Equations Using a Gini Covariance Matrix for the High-Dimensional Heavy-Tailed PVAR Model

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 614
Author(s):  
Jin Zou ◽  
Dong Han
Keyword(s):  
Granger Causality ◽  
Real Data ◽  
Vital Role ◽  
High Dimensional ◽  
Asymptotic Results ◽  
Vector Autoregressive ◽  
Matrix Estimation ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Gini Covariance

Gini covariance plays a vital role in analyzing the relationship between random variables with heavy-tailed distributions. In this papaer, with the existence of a finite second moment, we establish the Gini–Yule–Walker equation to estimate the transition matrix of high-dimensional periodic vector autoregressive (PVAR) processes, the asymptotic results of estimators have been established. We apply this method to study the Granger causality of the heavy-tailed PVAR process, and the results show that the robust transfer matrix estimation induces sign consistency in the value of Granger causality. Effectiveness of the proposed method is verified by both synthetic and real data.

scholarly journals Asymptotics for the Solutions to Defective Renewal Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Kaiyong Wang ◽  
Yang Chen ◽  
Zhongquan Tan
Keyword(s):  
Renewal Equations ◽  
Heavy Tailed ◽  
Distribution Class ◽  
Defective Renewal Equations

This paper investigates the defective renewal equations under the nonconvolution equivalent distribution class. The asymptotics of the solution to the defective renewal equations have been given for the heavy-tailed and light-tailed cases, respectively.

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.

scholarly journals A Method for Confidence Intervals of High Quantiles

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 70
Author(s):  
Mei Ling Huang ◽  
Xiang Raney-Yan
Keyword(s):  
Confidence Intervals ◽  
Computational Algorithm ◽  
Geometric Mean ◽  
Asymptotic Properties ◽  
Quantile Estimation ◽  
Heavy Tailed Distributions ◽  
High Quantiles ◽  
High Quantile ◽  
Heavy Tailed ◽  
Infinite Moments

The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. This paper proposes a new estimator for high quantiles based on the geometric mean. The new estimator has good asymptotic properties as well as it provides a computational algorithm for estimating confidence intervals of high quantiles. The new estimator avoids difficulties, improves efficiency and reduces bias. Comparisons of efficiencies and biases of the new estimator relative to existing estimators are studied. The theoretical are confirmed through Monte Carlo simulations. Finally, the applications on two real-world examples are provided.

scholarly journals Optimal Randomness for Stochastic Configuration Network (SCN) with Heavy-Tailed Distributions

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 56
Author(s):  
Haoyu Niu ◽  
Jiamin Wei ◽  
YangQuan Chen
Keyword(s):  
Research Work ◽  
Cauchy Distribution ◽  
Classification Model ◽  
Test Accuracy ◽  
Functional Link ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Improved Performance ◽  
Hidden Layer ◽  
Hidden Nodes

Stochastic Configuration Network (SCN) has a powerful capability for regression and classification analysis. Traditionally, it is quite challenging to correctly determine an appropriate architecture for a neural network so that the trained model can achieve excellent performance for both learning and generalization. Compared with the known randomized learning algorithms for single hidden layer feed-forward neural networks, such as Randomized Radial Basis Function (RBF) Networks and Random Vector Functional-link (RVFL), the SCN randomly assigns the input weights and biases of the hidden nodes in a supervisory mechanism. Since the parameters in the hidden layers are randomly generated in uniform distribution, hypothetically, there is optimal randomness. Heavy-tailed distribution has shown optimal randomness in an unknown environment for finding some targets. Therefore, in this research, the authors used heavy-tailed distributions to randomly initialize weights and biases to see if the new SCN models can achieve better performance than the original SCN. Heavy-tailed distributions, such as Lévy distribution, Cauchy distribution, and Weibull distribution, have been used. Since some mixed distributions show heavy-tailed properties, the mixed Gaussian and Laplace distributions were also studied in this research work. Experimental results showed improved performance for SCN with heavy-tailed distributions. For the regression model, SCN-Lévy, SCN-Mixture, SCN-Cauchy, and SCN-Weibull used less hidden nodes to achieve similar performance with SCN. For the classification model, SCN-Mixture, SCN-Lévy, and SCN-Cauchy have higher test accuracy of 91.5%, 91.7% and 92.4%, respectively. Both are higher than the test accuracy of the original SCN.

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