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AbstractRecent theoretical studies have shown that heavy-tails can emerge in stochastic optimization due to ‘multiplicative noise’, even under surprisingly simple settings, such as linear regression with Gaussian data. While these studies have uncovered several interesting phenomena, they consider conventional stochastic optimization problems, which exclude decentralized settings that naturally arise in modern machine learning applications. In this paper, we study the emergence of heavy-tails in decentralized stochastic gradient descent (DE-SGD), and investigate the effect of decentralization on the tail behavior. We first show that, when the loss function at each computational node is twice continuously differentiable and strongly convex outside a compact region, the law of the DE-SGD iterates converges to a distribution with polynomially decaying (heavy) tails. To have a more explicit control on the tail exponent, we then consider the case where the loss at each node is a quadratic function, and show that the tail-index can be estimated as a function of the step-size, batch-size, and the topological properties of the network of the computational nodes. Then, we provide theoretical and empirical results showing that DE-SGD has heavier tails than centralized SGD. We also compare DE-SGD to disconnected SGD where nodes distribute the data but do not communicate. Our theory uncovers an interesting interplay between the tails and the network structure: we identify two regimes of parameters (stepsize and network size), where DE-SGD can have lighter or heavier tails than disconnected SGD depending on the regime. Finally, to support our theoretical results, we provide numerical experiments conducted on linear regression and neural networks that suggest the heavier tails is correlated with better generalization in the decentralized setting. [ABSTRACT FROM AUTHOR]

Properties of the intervals between zero-crossings of non-Gaussian stable random processes are investigated. The classes of process considered are divided into those having short- and long-term memories, characterized by a coherence function that imbues dependence between a random variable measured at different times, which replaces the auto-correlation function, which exists only for the Gaussian case. For exponential coherence functions, all moments of probability densities for the intervals exist and a persistence parameter is determined that characterizes the rate of decay of the exponential tail of the probability densities, together with the full form of the density functions for selected values of the stability index. Results are verified with numerical simulation of a Cauchy process using a method independent of the theoretical development. For power-law coherence functions, the existence of moments of the distribution is conditional upon the indices characterizing the stable process and coherence function. The probability densities of the intervals have power-law tails depending on the product of these indices, with a logarithmic correction for specific combinations of these. An outer-scale or cut-off to power-law coherence functions regains a persistence parameter to the densities and the existence of all moments of the intervals. [ABSTRACT FROM AUTHOR]

AbstractThis paper introduces a novel family of distributions known as generalized scale mixtures of generalized normal distributions (GSMGN). These distributions incorporate two additional shape parameters that serve to regulate the shape and tails of the distribution. A finite mixture model based on this family is presented to address clustering heterogeneous data in the presence of leptokurtic and heavy-tailed outcomes. The estimation of the parameters of this model are obtained by developing an ECM-PLA ensemble algorithm which combine the profile likelihood approach (PLA) and the classical Expectation Conditional Maximization (ECM) algorithm, and the observed information matrix is obtained. The applicability of this new family and the numerical performance of the proposed methodology is discussed through simulated and real data examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the flexible interpretable gamma (FIG) distribution, with origins in Weibulisation, power weighting, and a stochastic representation. The FIG parameters have been verified graphically, mathematically, and through simulation as having separable roles in influencing the left tail, body, and right tail shape. The generalised gamma (GG) distribution has become a standard model for positive data in statistics due to its interpretable parameters and tractable equations. Although there are many generalised forms of the GG that can provide a better fit to data, none of them extend the GG so that the parameters are interpretable. We conduct simulation studies on the maximum likelihood estimates and respective sub-models of the FIG. Finally, we assess the flexibility of the FIG relative to existing models by applying the FIG model to hand grip strength and insurance loss data.

The probability that the sum of independent, centered, identically distributed, heavytailed random variables achieves a very large value is asymptotically equal to the probability that there exists a single summand equalling that value. We quantify the error in this approximation. We furthermore characterise the law of the individual summands, conditioned on the sum being large. [ABSTRACT FROM AUTHOR]

We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a characterization of regular variation on cones in [ 0 , ∞) d under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure. [ABSTRACT FROM AUTHOR]

I use extreme values theory and data on influenza mortality from the U.S. for 1900 to 2018 to estimate the tail risks of mortality. I find that the distribution for influenza mortality rates is heavy-tailed, which suggests that the tails of the mortality distribution are more informative than the events of high frequency (i.e., years of low mortality). I also discuss the implications of my estimates for risk management and pandemic planning.

Self-similarity is a prominent characteristic of natural images that can play a major role when it comes to their denoising, restoration or compression. In this paper, we propose a novel probabilistic model that is based on the concept of image patch similarity and applied to the problem of Single Image Super Resolution. Based on this model, we derive a Variational Bayes algorithm, which super resolves low-resolution images, where the assumed distribution for the quantified similarity between two image patches is heavy-tailed. Moreover, we prove mathematically that the proposed algorithm is both an extended and superior version of the probabilistic Non-Local Means (NLM). Its prime advantage remains though, which is that it requires no training. A comparison of the proposed approach with state-of-the-art methods, using various quantitative metrics shows that it is almost on par, for images depicting rural themes and in terms of the Structural Similarity Index (SSIM) with the best performing methods that rely on trained deep learning models. On the other hand, it is clearly inferior to them, for urban themed images and in terms of all metrics, especially for the Mean-Squared-Error (MSE). In addition, qualitative evaluation of the proposed approach is performed using the Perceptual Index metric, which has been introduced to better mimic the human perception of the image quality. This evaluation favors our approach when compared to the best performing method that requires no training, even if they perform equally in qualitative terms, reinforcing the argument that MSE is not always an accurate metric for image quality. [ABSTRACT FROM AUTHOR]

Zhao et al. (Nonlin. Dyn. 88, 477-487, 2017) presented the mutual information matrix (MIM) analysis for the study of nonlinear interactions in multivariate time series as an extension of Random Matrix Theory analysis. They considered the histogram estimation of mutual information based on Shannon entropy for discrete distributions. This paper is motivated by the latter, extending MIM analysis from a nonparametric and probabilistic discrete approach to a parametric and probabilistic continuous approach. Specifically, this paper presents the MIM based on Maximum Likelihood Estimators (MLEs) for flexible and tractable families of continuous multivariate distributions, called multivariate skew-elliptical families of distributions. This method focus on multivariate skew-Gaussian and skew-t distributions that allow modeling skewness and heavy-tails, respectively. Performance of the proposed methodology is illustrated by numerical results given by sinusoidal and vector autoregressive fractionally integrated moving-average models, and applied to a meteorological monitoring network data set. Results show that the consideration of skewness and heavy-tails in the transformed ozone time series produced some differences in the MIM estimations compared with those obtained by applying histogram estimations to transformed data. Given that mutual information index (MII) increases in line with the number of bins for the histogram estimator, the proposed methodology based on MLEs considered more robust estimators with respect to the histogram ones to determine the MII of multivariate time series. [ABSTRACT FROM AUTHOR]

We introduce a new family of multivariate distributions by taking the component-wise Tukey- h transformation of a random vector following a skew-normal distribution with an alternative parameterization. The proposed distribution is named the skew-normal-Tukey- h distribution and is an extension of the skew-normal distribution for handling heavy-tailed data. We compare this proposed distribution to the skew- t distribution, which is another extension of the skew-normal distribution for modeling tail-thickness, and demonstrate that when there are substantial differences in marginal kurtosis, the proposed distribution is more appropriate. Moreover, we derive many appealing stochastic properties of the proposed distribution and provide a methodology for the estimation of the parameters that can be applied to large dimensions. Using simulations, as well as a wine and a wind speed data application, we illustrate how to draw inferences based on the multivariate skew-normal-Tukey- h distribution. [ABSTRACT FROM AUTHOR]

Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable. [ABSTRACT FROM AUTHOR]

The Sibuya distribution is a discrete probability distribution on the positive integers which, while Poisson-compounding it, gives rise to the discrete-stable distribution of Steutel and van Harn. We first address the question of the discrete self-decomposability of Sibuya and Sibuya-related distributions. Discrete self-decomposable distributions arise as limit laws of pure-death branching processes with immigration, translating a balance between immigration events and systematic ageing and ultimate death of the immigrants at constant rate. Exploiting this fact, we design a new Luria–Delbrück-like model as an intertwining of a coexisting two-types (sensitive and mutant) population. In this model, a population of sensitive gently grows linearly with time. Mutants appear randomly at a rate proportional to the sensitive population size, very many at a time and with Sibuya-related distribution; each mutant is then immediately subject to random ageing and death upon appearance. The zero-set of the times free of mutants, when the sensitive population lacks immunity, is investigated using renewal theory. Finally, assuming each immigrant to die according to a critical binary branching processes, now with heavy-tailed extinction times, we observe that the local extinction events can become sparse, leading to a congestion of the mutants in the system. [ABSTRACT FROM AUTHOR]

In some applications of censored regression models, the distribution of the error terms departs significantly from normality, for instance, in the presence of heavy tails, skewness and/or atypical observation. In this paper we extend the censored linear regression model with normal errors to the case where the random errors follow a finite mixture of Student-t distributions. This approach allows us to model data with great flexibility, accommodating multimodality, heavy tails and also skewness depending on the structure of the mixture components. We develop an analytically tractable and efficient EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters, with standard errors as a by-product. The algorithm has closed-form expressions at the E-step, that rely on formulas for the mean and variance of the truncated Student-t distributions. The efficacy of the method is verified through the analysis of simulated and real datasets. The proposed algorithm and methods are implemented in the new R package CensMixReg. [ABSTRACT FROM AUTHOR]

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented.

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and elegant state-dependent efficient simulation algorithms have been developed in the literature over the last few years. We propose that by suitably decomposing these rare event probabilities into a dominant and further residual components, simpler state-independent importance sampling algorithms can be devised for each component resulting in composite unbiased estimators with desirable efficiency properties. When the increments have infinite variance, there is an added complexity in estimating the level crossing probabilities as even the well known zero-variance measures have an infinite expected termination time. We adapt our algorithms so that this expectation is finite while the estimators remain strongly efficient. Numerically, the proposed estimators perform at least as well, and sometimes substantially better than the existing state-dependent estimators in the literature.

In this paper, we address the problem of dynamic resource allocation in presence of job runtime uncertainty. We develop an execution delay model for runtime prediction, and design an adaptive stochastic allocation strategy, named Pareto Fractal Flow Predictor (PFFP). We conduct a comprehensive performance evaluation study of the PFFP strategy on real production traces, and compare it with other well-known non-clairvoyant strategies over two metrics. In order to choose the best strategy, we perform bi-objective analysis according to a degradation methodology. To analyze possible biasing results and negative effects of allowing a small portion of the problem instances with large deviation to dominate the conclusions, we present performance profiles of the strategies. We show that PFFP performs well in different scenarios with a variety of workloads and distributed resources. [ABSTRACT FROM AUTHOR]

In this work two video-on-demand (VoD) capacity models for H.264 video traces transmitted using 802.11g are proposed, one based on a self-similar traffic distribution and the other one based in the summation of a large number of Pareto distributed random variables. To ascertain the validity of using such modeling techniques a statistical analysis was performed where it was found that H.264 video traces exhibit self-similarity and heavy-tailed properties, as previous video formats that also use variable bit rate encoding. The models were evaluated against trace based simulations using ns-3 and results from hardware testbeds from other works. The model based on Pareto distributions gives a lower bound on a wide range of buffer sizes, while the model based on self-similarity provides a closer approximation to the user load when buffer size is high. The results show that the models can approximate the maximum user load for H.264 transmission on a local area VoD system and that they depend on the access point buffer size and the desired quality of service expressed as packet-loss probability. [ABSTRACT FROM AUTHOR]

Long-Range Dependence (LRD) and heavy-tailed distributions are ubiquitous in natural and socio-economic data. Such data can be self-similar whereby both LRD and heavy-tailed distributions contribute to the self-similarity as measured by the Hurst exponent. Some methods widely used in the physical sciences separately estimate these two parameters, which can lead to estimation bias. Those which do simultaneous estimation are based on frequentist methods such as Whittle’s approximate maximum likelihood estimator. Here we present a new and systematic Bayesian framework for the simultaneous inference of the LRD and heavy-tailed distribution parameters of a parametric ARFIMA model with non-Gaussian innovations. As innovations we use the α -stable and t -distributions which have power law tails. Our algorithm also provides parameter uncertainty estimates. We test our algorithm using synthetic data, and also data from the Geostationary Operational Environmental Satellite system (GOES) solar X-ray time series. These tests show that our algorithm is able to accurately and robustly estimate the LRD and heavy-tailed distribution parameters. [ABSTRACT FROM AUTHOR]

We consider the problem of the probabilistic quantification of dynamical systems that have heavy-tailed characteristics. These heavy-tailed features are associated with rare transient responses due to the occurrence of internal instabilities. Systems with these properties can be found in a variety of areas including mechanics, fluids, and waves. Here we develop a computational method, a probabilistic decomposition-synthesis technique, that takes into account the nature of internal instabilities to inexpensively determine the non-Gaussian probability density function for any arbitrary quantity of interest. Our approach relies on the decomposition of the statistics into a ‘non-extreme core’, typically Gaussian, and a heavy-tailed component. This decomposition is in full correspondence with a partition of the phase space into a ‘stable’ region where we have no internal instabilities, and a region where non-linear instabilities lead to rare transitions with high probability. We quantify the statistics in the stable region using a Gaussian approximation approach, while the non-Gaussian distribution associated with the intermittently unstable regions of phase space is inexpensively computed through order-reduction methods that take into account the strongly nonlinear character of the dynamics. The probabilistic information in the two domains is analytically synthesized through a total probability argument. The proposed approach allows for the accurate quantification of non-Gaussian tails at more than 10 standard deviations, at a fraction of the cost associated with the direct Monte-Carlo simulations. We demonstrate the probabilistic decomposition-synthesis method for rare events for two dynamical systems exhibiting extreme events: a two-degree-of-freedom system of nonlinearly coupled oscillators, and in a nonlinear envelope equation characterizing the propagation of unidirectional water waves. [ABSTRACT FROM AUTHOR]

We derive an analytical approximation to the probability distribution function (pdf) for the response of Mathieu׳s equation under parametric excitation by a random process with a spectrum peaked at the main resonant frequency, motivated by the problem of large amplitude ship roll resonance in random seas. The inclusion of random stochastic excitation renders the otherwise straightforward response to a system undergoing intermittent resonances : randomly occurring large amplitude bursts. Intermittent resonance occurs precisely when the random parametric excitation pushes the system into the instability region, causing an extreme magnitude response. As a result, the statistics are characterized by heavy-tails. We apply a recently developed mathematical technique, the probabilistic decomposition-synthesis method, to derive an analytical approximation to the non-Gaussian pdf of the response. We illustrate the validity of this analytical approximation through comparisons with Monte-Carlo simulations that demonstrate our result accurately captures the strong non-Gaussianinty of the response. [ABSTRACT FROM AUTHOR]

A weighted Gaussian approximation to tail product-limit process for Pareto-like distributions of randomly right-truncated data is provided and a new consistent and asymptotically normal estimator of the extreme value index is introduced. A simulation study is carried out to evaluate the finite sample behavior of the proposed estimator and compare it to that recently proposed by Gardes and Stupfler (TEST 24, 207-227, ). Also, a new approach of estimating extreme quantiles, under random right truncation, is derived and applied to a real dataset of lifetimes of automobile brake pads. [ABSTRACT FROM AUTHOR]

• Increasing spatial variability of rainfall determines heavier streamflow tails only beyond a certain increase threshold. • Small and elongated catchments are less resilient to increasing spatial variability of rainfall. • Daily records of rainfall and streamflow for a large set of catchments in Germany confirm simulations. Flow events with low frequency often cause severe damage, especially if their magnitudes are higher than suggested by historical observations. The heavier right tail of streamflow distribution indicates the increasing probability of high flows. In this paper, we investigate the role played by spatially variable rainfall in enhancing the tail heaviness of streamflow distributions. We synthetically generated a wide range of spatially variable rainfall inputs and fed them to a continuous probabilistic model of the catchment water transport to simulate streamflow in five German catchments with distinct properties in size and topography. Meanwhile, we used a comparable approach to analyze rainfall and runoff records from 175 German catchments. We identified the effects of spatially variable rainfall on the tails of streamflow distributions from both simulation scenarios and data analyses. Our results show that the tail of streamflow distribution becomes heavier with increasing spatial rainfall variability only beyond a certain threshold. This finding indicates the capability of catchments to buffer growing heterogeneities of rainfall, which we term catchment resilience to increasing spatial rainfall variability. The analyses suggest that the runoff routing through the river network controls this property. In fact, both small and elongated catchments are less resilient to increasing spatial rainfall variability due to their intrinsic runoff routing characteristics. We show the links between spatial rainfall characteristics and catchment geometry and the possible occurrence of high flows. The data analyses we performed on a large set of case studies confirm the simulation results and provide confidence for the transferability of these findings. [ABSTRACT FROM AUTHOR]

The right tail of streamflow distributions quantifies the occurrence probability of high flows, which play an important role in the dynamics of many eco-hydrological processes and eventually contribute to shape riverine environments. In this paper, the ability of a mechanistic analytical model for streamflow distributions to capture the statistical features of high flows has been investigated. The model couples a stochastic description of soil moisture dynamics with a simplified hydrologic response based on a catchment-scale storage–discharge relationship. Different types of relations between catchment water storage and discharge have been investigated, and alternative methods for parameter estimation have been compared using informal performance metrics and formal model selection criteria. The study highlights the pivotal role of non-linear storage–discharge relations in reproducing observed frequencies of high flows, and reveals the importance of analyzing the behavior of individual events for a reliable characterization of recession parameters. The emergence of heavy-tailed streamflow distributions is mechanistically linked to the degree of non-linearity of the catchment hydrologic response, with implications for the understanding of rivers’ flooding potential and related ecologic and morphological processes. [ABSTRACT FROM AUTHOR]

Modeling real data sets, even when we have some potential (as)symmetric models for the underlying data distribution, is always a very difficult task due to some uncontrollable perturbation factors. The analysis of different data sets from diverse areas of application, and in particular from statistical process control (SPC), leads us to notice that they usually exhibit moderate to strong asymmetry as well as light to heavy tails, which leads us to conclude that in most of the cases, fitting a normal distribution to the data is not the best option, despite of the simplicity and popularity of the Gaussian distribution. In this paper we consider a class of skew-normal models that include the normal distribution as a particular member. Some properties of the distributions belonging to this class are enhanced in order to motivate their use in applications. To monitor industrial processes some control charts for skew-normal and bivariate normal processes are developed, and their performance analyzed. An application with a real data set from a cork stopper’s process production is presented.

Von Bertalanffy growth models (VBGMs) have been used in several studies of age, growth and natural mortality. Assuming that the residuals about this growth model are normal is, however, questionable. Here, we assume that these residuals are heteroskedastic and follow a log-skew-t distribution, a flexible distribution that is asymmetric and heavy-tailed. We apply the proposed methodology to length-at-age data for the southern blue whiting (Micromesistius australis) collected from Chilean austral continental waters between 1997 and 2010. The estimates of the VBGM parameters were L∞ = 57.042 cm, K = 0.173 yr-1, t0 = -2.423 yr for males, and L∞ = 61.318 cm, K = 0.163 yr-1, t0 = -2.253 yr for females. The BIC criteria suggest that females grow significantly faster than males and that length-at-age for both sexes exhibits significant heteroskedasticity and asymmetry. [ABSTRACT FROM AUTHOR]

Although there are several software products dealing with the issue of simulating and estimating a stable distribution, SAS has no procedure for stable distributions. In this paper we propose two macros for estimating the parameters of a stable distribution using McCulloch method and Kogon-Williams method; further developments are required for implementing a procedure for estimating the parameters of a stable distribution using maximum likelihood method. [ABSTRACT FROM AUTHOR]

In this article, rare-event simulation for stochastic recurrence equations of the form X[sub n+1] = A[sub n+1]X[sub n] + B[sub n+1], X[sub 0] = 0 is studied, where {A[sub n]; n ≤ 1} and {B[sub n]; n ≤ 1} are independent sequences consisting of independent and identically distributed real-valued random variables. It is assumed that the tail of the distribution of B1 is regularly varying, whereas the distribution of A1 has a suitably light tail. The problem of efficient estimation, via simulation, of quantities such as P{X[sub n] > b} and P{sup[sub k≤b] X[sub k] > b} for large b and n is studied. Importance sampling strategies are investigated that provide unbiased estimators with bounded relative error as b and n tend to infinity. [ABSTRACT FROM AUTHOR]

CUSUM estimator is proposed for the change point in stochastic trend with heavy-tailed innovations. In order to avoid the outliers caused by heavy-tailed innovations, we also construct a truncating CUSUM estimator. Results in this paper show that the CUSUM estimators are consistent. Simulations demonstrate that the truncating estimator behaves better for the heavy-tailed innovations. [ABSTRACT FROM AUTHOR]

Abstract: Some foragers have movement patterns that can be approximated by Lévy walks whilst others may be better represented as composite Brownian walks. Many attempts have been made to interpret these movement patterns in terms of optimal searching strategies for the location of randomly and sparsely distributed targets. Here it is shown that the relative merits of Lévy walk and composite Brownian walk searches are sensitively dependent upon target encounter dynamics which set the initial conditions for an extensive search. It is suggested these initial conditions are determined, at least in part, by the competing demands of harvesting and safety from predation. In accordance with observations, it is shown that Lévy walks are expected in tritrophic systems and where intraguild predation operates. Composite Brownian walks, on the other hand, are found to be advantageous when the risk of predation is low. Despite having fundamentally different properties, Lévy walks and composite Brownian walks can therefore compete a priori as possible models of animal movements. Throughout, attention is focused on searching for randomly and sparsely distributed resources that are not depleted or rejected once located but instead remain targets for future searches. We re-evaluate and overturn the widely held belief that in numerical simulations this ‘non-destructive’ searching scenario can faithfully and consistently represent destructive searching for patchily distributed resources, i.e. for resources that tend to occur in clusters rather than in isolation. [Copyright &y& Elsevier]

We derive two scalability models for high-performance distributed environments using low-cost, low-performance components. On the first model, we approximate the tail of the probability distribution by using the Pareto probability distribution and then build a stochastic model to determine the maximum user-load per server according to certain quality of service parameters. On the second model, we use a probabilistic measure to determine the ratio of computing servers to storage servers and thus complete the performance model. The data of execution traces obtained from the testbed is analyzed. Hurst parameter is estimated using the Abry–Veitch estimator and distribution parameters are estimated for different probability distributions. Models based on the Pareto and Gamma probability distributions are developed and used with the data that was analyzed. There seems to be a good agreement between our models, the experimental execution traces, and simulations of HP computing environments. Specifically, the Pareto Fractal Flow model is compared with other models based on the Gamma and Gaussian distributions, such as the FGN and M/G/∞, and it seems to make better predictions under our experimental conditions. Results are presented comparing analytical models with simulation results. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we first analyze the stylized facts of electricity prices, in particular, the extreme volatility and price spikes which lead to heavy-tailed distributions of price changes. Then we calibrate Markov regime-switching (MRS) models with heavy-tailed components and show that they adequately address the aforementioned characteristics. Contrary to the common belief that electricity price models ‘should be built on log-prices’, we find evidence that modeling the prices themselves is more beneficial and methodologically sound, at least in case of MRS models. [ABSTRACT FROM AUTHOR]

We consider the problem of maximum likelihood estimation of the parameters of the Pareto Type II (Lomax) distribution. We show that in certain parametrization and after modification of the parameter space to include exponential distribution as a special case, the MLEs of parameters always exist. Moreover, the MLEs have a non standard asymptotic distribution in the exponential case due to the lack of regularity. Further, we develop a likelihood ratio test for exponentiality versus Pareto II distribution. We emphasize that this problem is non standard, and the limiting null distribution of the deviance statistic in not chi-square. We derive relevant asymptotic theory as well as a convenient computational formula for the critical values for the test. An empirical power study and power comparisons with other tests are also provided. A problem from climatology involving precipitation data from hundreds of meteorological stations across North America provides a motivation for and an illustration of the new test. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result for the asymptotic distribution of the sample average value-at-risk (AVaR) under certain regularity assumptions. The asymptotic distribution can be used to derive asymptotic confidence intervals when AVaRϵ(X) is calculated by the Monte Carlo method which is adopted in many risk management systems. We study the effect of the tail behavior of the random variable X on the convergence rate and the improvement of a tail truncation method. [ABSTRACT FROM AUTHOR]

In this paper, we provide a stable limit theorem for the asymptotic distribution of the sample average value-at-risk when the distribution of the underlying random variable X describing portfolio returns is heavy-tailed. We illustrate the convergence rate in the limit theorem assuming that X has a stable Paretian distribution and Student's t distribution. [ABSTRACT FROM AUTHOR]

Abstract: We solve a problem of non-convex stochastic optimisation with help of simulated annealing of Lévy flights of a variable stability index. The search of the ground state of an unknown potential is non-local due to big jumps of the Levy flights process. The convergence to the ground state is fast due to a polynomial decrease rate of the temperature. [Copyright &y& Elsevier]

A broad range of statistical processes is characterized by the generalized Gaussian statistics. For instance, the Gaussian and Laplacian probability density functions are special cases of generalized Gaussian statistics. Moreover, the linear and median filtering structures are statistically related to the maximum likelihood estimates of location under Gaussian and Laplacian statistics, respectively. In this paper, we investigate the well-established statistical relationship between Gaussian and Cauchy distributions, showing that the random variable formed as the ratio of two independent Gaussian distributed random variables is Cauchy distributed. We also note that the Cauchy distribution is a member of the generalized Cauchy distribution family. Recently proposed myriad filtering is based on the maximum likelihood estimate of location under Cauchy statistics. An analogous relationship is formed here for the Laplacian statistics, as the ratio of Laplacian statistics yields the distribution referred here to as the Meridian. Interestingly, the Meridian distribution is also a member of the generalized Cauchy family. The maximum likelihood estimate under the obtained statistics is analyzed. Motivated by the maximum likelihood estimate under meridian statistics, meridian filtering is proposed. The analysis presented here indicates that the proposed filtering structure exhibits characteristics more robust than that of median and myriad filtering structures. The statistical and deterministic properties essential to signal processing applications of the meridian filter are given. The meridian filtering structure is extended to admit real-valued weights utilizing the sign coupling approach. Finally, simulations are performed to evaluate and compare the proposed meridian filtering structure performance to those of linear, median, and myriad filtering. [ABSTRACT FROM AUTHOR]

In this paper we present a method for performing regression with stable disturbances. The method of maximum likelihood is used to estimate both distribution and regression parameters. Our approach utilises a numerical integration procedure to calculate the stable density, followed by sequential quadratic programming optimisation procedures to obtain estimates and standard errors. A theoretical justification for the use of stable law regression is given followed by two real world practical examples of the method. First, we fit the stable law multiple regression model to housing price data and examine how the results differ from normal linear regression. Second, we calculate the beta coefficients for 26 companies from the Financial Times Ordinary Shares Index. [ABSTRACT FROM AUTHOR]

This article considers the problem of a mean change-point in heavy-tailed dependent observations. A method of change-point estimation by truncating initial process is proposed, which can weaken the affection of outliers. In the infinite variance case, we obtained a generalization Hájek-Rényi type inequality. Consistency and the rate of convergence for the estimated change-point are also established. The results of a simulation study support validity of our method. [ABSTRACT FROM AUTHOR]

This paper addresses issues that arise in copyright protection systems of digital images, which employ blind watermark verification structures in the discrete cosine transform (DCT) domain. First, we observe that statistical distributions with heavy algebraic tails, such as the alpha-stable family, are in many cases more accurate modeling tools for the DCT coefficients of JPEG-analyzed images than families with exponential tails such as the generalized Gaussian. Motivated by our modeling results, we then design a new processor for blind watermark detection using the Cauchy member of the alpha-stable family. The Cauchy distribution is chosen because it is the only non-Gaussian symmetric alpha-stable distribution that exists in closed form and also because it leads to the design of a nearly optimum detector with robust detection performance. We analyze the performance of the new detector in terms of the associated probabilities of detection and false alarm and we compare it to the performance of the generalized Gaussian detector by performing experiments with various test images. [ABSTRACT FROM PUBLISHER]

Embrechts and Veraverbeke[2] investigated the renewal risk model and gave a tail equivalence relationship of the ruin probabilities ψ( x) under the assumption that the claim size is heavy-tailed, which is regarded as a classical result in the context of extremal value theory. In this note we extend this result to the delayed renewal risk model. [ABSTRACT FROM AUTHOR]