scholarly journals Fourier transform MCMC, heavy-tailed distributions, and geometric ergodicity

2021 ◽  
Vol 181 ◽  
pp. 351-363
Author(s):  
Denis Belomestny ◽  
Leonid Iosipoi
Keyword(s):  
Fourier Transform ◽  
Geometric Ergodicity ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

scholarly journals Ruin Probabilities and Complex Analysis

2021 ◽  
Author(s):  
Andrew Leung
Keyword(s):  
Fourier Transform ◽  
Continuous Time ◽  
Complex Analysis ◽  
Inverse Fourier Transform ◽  
Ruin Probabilities ◽  
Heavy Tailed Distributions ◽  
Complex Functions ◽  
Heavy Tailed ◽  
The Fourier Transform ◽  
The Relationship

This paper considers the solution of the equations for ruin probabilities in infinite continuous time. Using the Fourier Transform and certain results from the theory of complex functions, these solutions are obtained as com- plex integrals in a form which may be evaluated numerically by means of the inverse Fourier Transform. In addition the relationship between the re- sults obtained for the continuous time cases, and those in the literature, are compared. Closed form ruin probabilities for the heavy tailed distributions: mixed exponential; Gamma (including Erlang); Lognormal; Weillbull; and Pareto, are derived as a result (or computed to any degree of accuracy, and without the use of simulations).

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.

scholarly journals Computation of Compound Distributions I: Aliasing Errors and Exponential Tilting

1999 ◽  
Vol 29 (2) ◽  
pp. 197-214 ◽  
Author(s):  
Rudolf Grübel ◽  
Renate Hermesmeier
Keyword(s):  
Numerical Evaluation ◽  
Change Of Measure ◽  
Exponential Tilting ◽  
Transform Methods ◽  
Compound Distributions ◽  
Heavy Tailed Distributions ◽  
Panjer Recursion ◽  
Heavy Tailed ◽  
Suitable Change

AbstractNumerical evaluation of compound distributions is one of the central numerical tasks in insurance mathematics. Two widely used techniques are Panjer recursion and transform methods. Many authors have pointed out that aliasing errors imply the need to consider the whole distribution if transform methods are used, a potential drawback especially for heavy-tailed distributions. We investigate the magnitude of aliasing errors and show that this problem can be solved by a suitable change of measure.

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