###### multivariate

- Despite this evidence, non-normal multivariate modelling of asset returns does not appear to play an important role in asset management or risk management because of the complexity of estimating multivariate non-normal distributions from market return data. (repec.org)
- In this paper, we present a new subclass of generalized elliptical distributions for asset returns that is sufficiently user friendly, so that it can be utilized by asset managers and risk managers for modelling multivariate non-normal distributions of asset returns. (repec.org)
- Her current focus is on multivariate risk such as fat tail distribution and copula. (edu.au)
- Cumulative distribution functions are also used to specify the distribution of multivariate random variables . (wikipedia.org)

###### Poisson

- The proper use of tables of the binomial and Poisson distributions depends upon this convention. (wikipedia.org)
- The variance is very close to the mean, so the Poisson distribution for the entire data set seems like a good starting point. (stackexchange.com)

###### empirical

- In the study of asset returns, the preponderance of empirical evidence finds that return distributions are not normally distributed. (repec.org)
- Finally, as an illustration, we conduct an empirical analysis to explore the performance of different models representing the distributions of log-returns for the S&P500 and DAX indexes. (springer.com)
- The law of the iterated logarithm for normalized empirical distribution function. (springer.com)
- Laws of the iterated logarithm in the tails for weighted uniform empirical processes. (springer.com)
- On the relationship between the almost sure stability of weighted empirical distributions and sums of order statistics. (springer.com)
- Characterisation of the class of limiting distributions of empirical point processes of exceedances (EPPEs), necessary and sufficient conditions for the complete convergence of an EPPE to a particular limit law. (mdx.ac.uk)
- Unfortunately, the empirical detection and characterization of power laws is made difficult by the large fluctuations that occur in the tail of the distribution. (scribd.com)

###### stochastic

- A simple stochastic mechanism that produces exact and approximate power-law distributions is presented. (scirp.org)
- These applications motivate new stochastic averaging results for systems with fast processes driven by heavy-tailed noise. (gatech.edu)
- Therefore, this study aims to develop a flexible four-variate joint distribution function of the regional stochastic nature of drought. (springer.com)

###### Pareto

- Newman, M.E.J. (2005) Power Laws, Pareto Distributions and Zipfs Law, Contemporary Physics. (scirp.org)
- size of cities Source: Denise Pumain, Scaling Laws and Urban Distributions , 2003 IRDM WS 2005 Example Pareto Distribution size of file requests Source: Mark Crovella et al. (zanran.com)

###### Laplace-Gaussian

- Modelling and predicting market risk with Laplace-Gaussian mixture distributions ," Applied Financial Economics , Taylor & Francis Journals, vol. 16(15), pages 1145-1162. (repec.org)
- Modeling and predicting market risk with Laplace-Gaussian mixture distributions ," CFS Working Paper Series 2005/11, Center for Financial Studies (CFS). (repec.org)

###### symmetric

- The Uniform distribution is, not coincidentally, symmetric. (statisticsblog.com)
- For symmetric distributions like the Normal, we only consider positive values. (statisticsblog.com)
- As these drought variables showed significant dependence properties and followed different marginal distributions, we employed and compared six four-variate symmetric and asymmetric Archimedean copulas (i.e. (springer.com)

###### Prevalence

- Here, we estimate the prevalence and direction of black-swan events (heavy-tailed process noise) in 609 animal populations after accounting for population dynamics (productivity, density dependence, and typical stochasticity). (pnas.org)

###### Asmussen

- Asmussen, S., Møller, J.R.: Tail asymptotics for M / G /1 type queueing processes with subexponential increments. (springer.com)

###### sums

- A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. (springer.com)
- Asymptotic distribution of trimmed sums. (springer.com)
- Limit Distributions for Sums of Independent Random Vectors. (researchandmarkets.com)
- Limit Distributions for Sums of Independent Random Vectors is a comprehensive reference that provides an up-to-date survey of the state of the art in this important research area. (researchandmarkets.com)

###### behaviour

- For the distribution we present, which we call the multi-tail generalized elliptical distribution, we (1) derive the densities using results of the theory of generalized elliptical distributions and (2) introduce a function, which we label the tail function, to describe their tail behaviour. (repec.org)
- We test the model on German stock returns and find that (1) the multi-tail model introduced in the paper significantly outperforms the classical elliptical model and (2) the hypothesis of homogeneous tail behaviour can be rejected. (repec.org)
- In this study, we investigate the performance of different parametric models with stable and tempered stable distributions for capturing the tail behaviour of log-returns (financial asset returns). (springer.com)
- Alsmeyer, G., Sbignev, M.: On the tail behaviour of the supremum of a random walk defined on a Markov chain. (springer.com)

###### Bernoulli

- Think of it this way: each sample of 4 points gives us one Bernoulli trial from a single distribution, which returns a 0 or 1. (statisticsblog.com)

###### heavy

- Randomly sampling these functions with a radially uniform sampling scheme produces heavy-tailed distributions. (scirp.org)
- Crovella, M. E., Taqqu, M. S., and Bestavros, A. Heavy - tailed probability distributions in the World Wide Web . (zanran.com)
- A primer on heavy - tailed distributions . (zanran.com)
- and A. Bestavros, " Heavy - tailed probability distributions in the world wide web , " A Practical Guide To Heavy Tails , Chapman & Hall, New York, pp. 3-26, 1998. (zanran.com)
- Equilibrium rm size distributions are heavy tailed and skewed to the right. (repec.org)
- Indeed, since most of the studies which employ the Laplace distribution are concerned with modelling heavy tailed patterns, the modern class of possible alternatives is way broader than just testing the Laplace vs. normal distribution. (repec.org)
- Heavy tailed distributions have been shown to be consistent with data in a variety of systems with multiple time scales. (gatech.edu)
- For example, correlated additive and multiplicative (CAM) Gaussian noise, with infinite variance or heavy tails in certain parameter regimes, has received increased attention in the context of atmosphere and ocean dynamics. (gatech.edu)
- We develop these results for the case of alpha-stable noise, and discuss open problems for identifying appropriate heavy tailed distributions for these multiple scale systems. (gatech.edu)
- Heavy Tails in Theory and Practice. (researchandmarkets.com)
- Novak S.Y. (2014) Lower bounds to the accuracy of inference on heavy tails. (mdx.ac.uk)
- At MSO4110 module students learn advanced methods of evaluating measures of financial risk from dependent heavy-tailed data. (mdx.ac.uk)
- Estimation of the covariance structure of heavy-tailed distributions. (google.com)
- Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. (springer.com)
- In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. (springer.com)
- Foss, S., Zachary, S.: Asymptotics for the maximum of a modulated random walk with heavy-tailed increments. (springer.com)
- Black-swan events manifest primarily as population die-offs and crashes (86%) rather than unexpected increases, and ignoring heavy-tailed process noise leads to an underestimate in the magnitude of population crashes. (pnas.org)
- We suggest modelers consider heavy-tailed, downward-skewed probability distributions, such as the skewed Student t used here, when making forecasts of population abundance. (pnas.org)
- Our results demonstrate the importance of both modeling heavy-tailed downward events in populations, and developing conservation strategies that are robust to ecological surprises. (pnas.org)

###### parameters

- Blue Mussels Mytilus edulis are the main food source for Longtailed Ducks in the southern Baltic Sea and here we have used both modelled and measured raw data on Mytilus abundances, patchiness and various bathymetric parameters to study Long-tailed Duck habitat preferences. (lu.se)
- In particular, standard methods such as least-squares fitting are known to produce systematically biased estimates of parameters for power-law distributions and should not be used in most circumstances. (scribd.com)

###### uniform

- For example, the red dots are averages from a Uniform distribution. (statisticsblog.com)
- The more trials you do, the closer results from the Uniform will cluster around 0.5, which is the "true" Obesity value for this distribution. (statisticsblog.com)
- On the other hand, I'd like to see a greater distance between the Uniform and Beta(0.01, 0.01) distribution, as the latter is an extreme case of small tails. (statisticsblog.com)
- Extra special karma points to the first person who comes up with a distribution whose Obesity index is between the Uniform and Normal, and that isn't a variant of one I already tested. (statisticsblog.com)

###### discrete probability

- From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part. (wikipedia.org)

###### Explanation

- Excellent explanation of Normal vs. Fat-tailed Distributions . (highscalability.com)

###### Statistical

- Scientists have learned many things from observation such as these are not well characterized by their aver- of the statistical distributions of experimental quantities ages. (scribd.com)

###### estimation

- Novak S.Y. (2013) Lower bounds to the accuracy of tail index estimation. (mdx.ac.uk)
- The estimation procedure we have followed to detect the presence of power law tails in the distributions of the economic agents, is the one proposed by Clementi et al. (springer.com)

###### depends

- The heaviness of the tail depends on one industry-level parameter. (repec.org)

###### data

- Comparison of some tests of fit for the Laplace distribution ," Computational Statistics & Data Analysis , Elsevier, vol. 52(12), pages 5338-5343, August. (repec.org)
- To prevent this attack, the data in RAM is overwritten by random data when shutting down Tails. (ghacks.net)
- Each of the data sets has been conjectured previously to follow a power-law distribution. (scribd.com)

###### stable distribution

- Bergström, H.: On distribution functions with a limiting stable distribution function. (springer.com)

###### respectively

- In double logarithmic plot of L (distance of a base to the next base, x-axis) vs F (frequencies of a base at L, y-axis), each value of four bases was expressed in y = ae -bx at L = 1 ~ 15, and y = Ux + W (power-law-tail) at L = more than 16 bases, respectively, in a single-strand of DNA. (ccsenet.org)

###### asymptotic

- The asymptotic distribution of the trimmed mean. (springer.com)
- Arndt, K.: Asymptotic properties of the distribution of the supremum of a random walk on a Markov chain. (springer.com)

###### Probability Theory

- The authors develop the central limit theory in detail, starting with the basic constructions of modern probability theory, then developing the fundamental tools of infinitely divisible distributions and regular variation. (researchandmarkets.com)
- In probability theory and statistics , the cumulative distribution function ( CDF ) of a real-valued random variable X , or just distribution function of X , evaluated at x , is the probability that X will take a value less than or equal to x . (wikipedia.org)

###### moments

- On the influence of moments on the rate of convergence to the normal distribution. (springer.com)

###### results

- The dots in the graph represent the mean results from 8, 16, 32, and so on, up to 4096 trials from each of the distributions I tested. (statisticsblog.com)
- These results indicated that the power-law-tail would be specific for the complexity of organisms in individual genome, and might be a new index for cells. (ccsenet.org)

###### properties

- Over the last decade there has been a marked interest in a Laplace distribution and its properties and generalizations, especially in the framework of financial applications. (repec.org)
- Every function with these four properties is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable. (wikipedia.org)

###### different

- The graph at the top of this post shows how the Obesity index converges for different distributions. (statisticsblog.com)
- The graph gives a feel for how many trials would be needed to distinguish between different distributions based on their Obesity index. (statisticsblog.com)
- Installing Tails is not quite as simple as installing other distributions, and the method is a little different depending on if you're using Windows, Ubuntu/Debian/Mint or other forms of GNU/Linux. (ghacks.net)

###### size

- R&D Behavior and the Emergence of Fat Tailed Firm Size Distributions ," MERIT Working Papers 056, United Nations University - Maastricht Economic and Social Research Institute on Innovation and Technology (MERIT). (repec.org)
- On the Evolution of the Firm Size Distribution: Facts and Theory ," American Economic Review , American Economic Association, vol. 93(4), pages 1075-1090, September. (repec.org)
- On the Evolution of the Firm Size Distribution: Facts and Theory ," CEPR Discussion Papers 3045, C.E.P.R. Discussion Papers. (repec.org)
- Thus, the power-law-tail of the genomic DNA should be come from the structural features of the cells, i.e., the size, the GC-content and other characteristics of the genomic DNA. (ccsenet.org)
- Distribution-Free, Size Adaptive Submatrix Detection with Acceleration. (google.com)

###### Models

- Mitzenmacher, M. (2003) A Brief History of Generative Models for Power Law and Lognormal Distributions. (scirp.org)

###### maximum

- Based on the fact that one cannot be assured of observing population maximum lifespans in finite samples, in previous work, we constructed and validated several tests of difference in the upper parts of lifespan distributions between a treatment group and a control group by testing whether the probabilities that observations are above some threshold defining 'old' or being in the tail of the survival distribution are equal in the two groups. (biomedcentral.com)
- Foss, S., Zachary, S.: The maximum on a random time interval of a random walk with long-tailed increments and negative drift. (springer.com)

###### continuous

- In the case of a continuous distribution , it gives the area under the probability density function from minus infinity to x . (wikipedia.org)
- Every cumulative distribution function F is non-decreasing and right-continuous , which makes it a càdlàg function. (wikipedia.org)

###### specific

- It's no secret that using GNU/Linux is generally safer than using Microsoft Windows, however, there are specific distributions of GNU/Linux that are even more focused on privacy and safety than the usual distros like Ubuntu. (ghacks.net)
- This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution . (wikipedia.org)

###### negative

- Let ( X n ) n≥1 be a sequence of independent non-negative random variables from a common distribution function F with a regiilarly varying upper tail. (springer.com)

###### main

- The Long-tailed Duck Clangula hyemalis is the main offshore wintering seaduck species in the Baltic region, although numbers have declined steeply since the early 1990s. (lu.se)

###### random

- x n be a sequence of independent random variables with the common distribution F. Suppose E x k =0 and that F belongs to the domain of attraction of the normal distribution. (springer.com)

###### function

- It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions . (wikipedia.org)
- The function f is equal to the derivative of F almost everywhere , and it is called the probability density function of the distribution of X . (wikipedia.org)

###### found

- For full instructions on how to install Tails, check out the guide found here . (ghacks.net)

###### power-law distribu

- the lifetimes of intellectual artifacts follow a power law distribution. (highscalability.com)

###### sample

- Step 1: Sample four times from a distribution. (statisticsblog.com)
- i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution). (gerardnico.com)
- Nevertheless, there is appeal to using the full continuity of information in the upper tails of the sample distribution, and colleagues have recently suggested to us that a limitation of the Wang-Allison tests is that they only treat individual lifespans as being above or below some threshold defining 'old' or being in the tail of the survival distribution. (biomedcentral.com)

###### functions

- Esséen, C.G.: Fourier analysis of distribution functions. (springer.com)

###### shown

- The developed test statistic is shown to asymptotically follow a [chi]2-distribution with two degrees of freedom. (repec.org)

###### upper

- 5 ] constructed and validated several tests (hereafter, the 'Wang-Allison tests' ) of differences in the upper parts of lifespan distributions by building on the work of Redden et al. (biomedcentral.com)

###### possible

- I've done it this way as part of my Grand Master Plan to map every possible distribution based on how it performs in a variety of tail indices. (statisticsblog.com)
- Tails is ideal for keeping yourself and your activities hidden as previously stated, which is absolutely perfect for those who want to do their best to have 'big brother' unable to spy on their activities, as much as possible. (ghacks.net)

###### example

- Edward Snowden for example, was using Tails to communicate with Glen Greenwald and others, when he released his famous leaks of classified information. (ghacks.net)

###### paper

- In this paper we propose a new test of fit for a Laplace distribution against deviations with heavier tails than that of the reference Laplace distribution. (repec.org)
- In my never ending quest to find the perfect measure of tail fatness , I ran across this recent paper by Cooke, Nieboer, and Misiewicz. (statisticsblog.com)