scholarly journals Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions

2014 ◽  
Vol 05 (13) ◽  
pp. 2050-2056
Author(s):  
Frederic von Wegner
Keyword(s):  
Power Law ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

From the central limit theorem to heavy-tailed distributions

2003 ◽  
Vol 40 (3) ◽  
pp. 803-806 ◽  
Author(s):  
Jinwen Chen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Power Law ◽  
Power Law Distribution ◽  
The Central Limit Theorem ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Random Indices ◽  
Power Law Distributions

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

Heavy-tailed distributions for building stock data

2018 ◽  
Vol 46 (7) ◽  
pp. 1281-1296 ◽  
Author(s):  
Patrick Erik Bradley ◽  
Martin Behnisch
Keyword(s):  
Power Law ◽  
Building Stock ◽  
Underlying Distribution ◽  
Large Cities ◽  
Statistical Framework ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed Distribution ◽  
Event Histories ◽  
Heavy Tailed ◽  
Log Normal

The question of inferring the owner of a set of building stocks (e.g. from which country the buildings are taken) from building-related quantities like number of buildings or types of building event histories necessitates the knowledge of their distributions in order to compare them. If the distribution function is a power law, then a version of the 80/20 rule can be applied to describe the variable. This distribution is an example of a heavy-tailed distribution; another example is the log-normal distribution. Heavy-tailed distributions have the property that studying the effects of the few large values already yields most of the overall effect of the whole quantity. For example, if reducing the CO2 emissions of the buildings of a country is the issue, then in case of a heavy-tailed distribution, only the effects of the relatively few large cities need to be considered. It is shown that the number of buildings in German municipalities or counties or the number of building-related event histories of a certain vanished building stock follow a heavy-tailed distribution and give evidence for the type of underlying distribution. The methodology used is a recent statistical framework for discerning power law and other heavy-tailed distributions in empirical data.

An analysis of power law distributions and tipping points during the global financial crisis

2018 ◽  
Vol 13 (1) ◽  
pp. 80-91 ◽  
Author(s):  
Yifei Li ◽  
Lei Shi ◽  
Neil Allan ◽  
John Evans
Keyword(s):  
Financial Crisis ◽  
Power Law ◽  
Global Financial Crisis ◽  
Operational Risk ◽  
Tipping Point ◽  
Power Law Distribution ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
The Global Financial Crisis ◽  
Power Law Distributions

AbstractHeavy-tailed distributions have been observed for various financial risks and papers have observed that these heavy-tailed distributions are power law distributions. The breakdown of a power law distribution is also seen as an indicator of a tipping point being reached and a system then moves from stability through instability to a new equilibrium. In this paper, we analyse the distribution of operational risk losses in US banks, credit defaults in US corporates and market risk events in the US during the global financial crisis (GFC). We conclude that market risk and credit risk do not follow a power law distribution, and even though operational risk follows a power law distribution, there is a better distribution fit for operational risk. We also conclude that whilst there is evidence that credit defaults and market risks did reach a tipping point, operational risk losses did not. We conclude that the government intervention in the banking system during the GFC was a possible cause of banks avoiding a tipping point.

scholarly journals Heavy-tailed distributions, correlations, kurtosis and Taylor’s Law of fluctuation scaling

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200610
Author(s):  
Joel E. Cohen ◽  
Richard A. Davis ◽  
Gennady Samorodnitsky
Keyword(s):  
Power Law ◽  
Statistical Models ◽  
Natural Conditions ◽  
Heavy Tailed Distributions ◽  
Sample Correlation ◽  
Taylor’S Law ◽  
Heavy Tailed ◽  
Fluctuation Scaling ◽  
Random Limit ◽  
Asymptotic Scaling

Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44 , 2089–2097; p. 2091) speculated that ‘the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·’. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor’s Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances.

From the central limit theorem to heavy-tailed distributions

2003 ◽  
Vol 40 (03) ◽  
pp. 803-806
Author(s):  
Jinwen Chen
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Power Law ◽  
Power Law Distribution ◽  
The Central Limit Theorem ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Random Indices ◽  
Power Law Distributions

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.

scholarly journals Large-Scale, Wavelet-Based Analysis of Lysosomal Trajectories and Co-Movements of Lysosomes with Nanoparticle Cargos

Cells ◽  
2022 ◽  
Vol 11 (2) ◽  
pp. 270
Author(s):  
Konstantin Polev ◽  
Diana V. Kolygina ◽  
Kristiana Kandere-Grzybowska ◽  
Bartosz A. Grzybowski
Keyword(s):  
Power Law ◽  
Large Scale ◽  
Cell Types ◽  
Potential Models ◽  
Movement Trajectories ◽  
Heavy Tailed Distributions ◽  
Nanoparticle Aggregates ◽  
Heavy Tailed ◽  
Acidic Organelles ◽  
Cancerous Cells

Lysosomes—that is, acidic organelles known for degradation/recycling—move through the cytoplasm alternating between bursts of active transport and short, diffusive motions or even pauses. While their mobility is essential for lysosomes’ fusogenic and non-fusogenic interactions with target organelles, their movements have not been characterized in adequate detail. Here, large-scale statistical analysis of lysosomal movement trajectories reveals that lysosome trajectories in all examined cell types—both cancer and noncancerous ones—are superdiffusive and characterized by heavy-tailed distributions of run and flight lengths. Consideration of Akaike weights for various potential models (lognormal, power law, truncated power law, stretched exponential, and exponential) indicates that the experimental data are best described by the lognormal distribution, which, in turn, can be related to one of the space-search strategies particularly effective when “thorough” search needs to balance search for rare target(s) (organelles). In addition, automated, wavelet-based analysis allows for co-tracking the motions of lysosomes and the cargos they carry—particularly the nanoparticle aggregates known to cause selective lysosome disruption in cancerous cells. The methods we describe here could help study nanoparticle assemblies, viruses, and other objects transported inside various vesicle types, as well as coordinated movements of organelles/particles in the cytoplasm. Custom-written code that includes integrated workflow for our analyses is made available for academic use.

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.

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