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## Limiting behavior of delayed sums under a non-identically distribution setup

Pingyan,Chen
Fonte: Academia Brasileira de Ciências Publicador: Academia Brasileira de Ciências
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/12/2008 Português
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We present an accurate description the limiting behavior of delayed sums under a non-identically distribution setup, and deduce Chover-type laws of the iterated logarithm for them. These complement and extend the results of Vasudeva and Divanji (Theory of Probability and its Applications, 37 (1992), 534-542).

## Ship Detection in SAR Image Based on the Alpha-stable Distribution

Wang, Changcheng; Liao, Mingsheng; Li, Xiaofeng
Fonte: Molecular Diversity Preservation International (MDPI) Publicador: Molecular Diversity Preservation International (MDPI)
Tipo: Artigo de Revista Científica
Publicado em 22/08/2008 Português
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This paper describes an improved Constant False Alarm Rate (CFAR) ship detection algorithm in spaceborne synthetic aperture radar (SAR) image based on Alpha-stable distribution model. Typically, the CFAR algorithm uses the Gaussian distribution model to describe statistical characteristics of a SAR image background clutter. However, the Gaussian distribution is only valid for multilook SAR images when several radar looks are averaged. As sea clutter in SAR images shows spiky or heavy-tailed characteristics, the Gaussian distribution often fails to describe background sea clutter. In this study, we replace the Gaussian distribution with the Alpha-stable distribution, which is widely used in impulsive or spiky signal processing, to describe the background sea clutter in SAR images. In our proposed algorithm, an initial step for detecting possible ship targets is employed. Then, similar to the typical two-parameter CFAR algorithm, a local process is applied to the pixel identified as possible target. A RADARSAT-1 image is used to validate this Alpha-stable distribution based algorithm. Meanwhile, known ship location data during the time of RADARSAT-1 SAR image acquisition is used to validate ship detection results. Validation results show improvements of the new CFAR algorithm based on the Alpha-stable distribution over the CFAR algorithm based on the Gaussian distribution.

## Statistical description of the error on wind power forecasts via a Lévy a-stable distribution

BRUNINX, Kenneth; DELARUE, Erik; D’HAESELEER, William
Fonte: Instituto Universitário Europeu Publicador: Instituto Universitário Europeu
Tipo: Trabalho em Andamento Formato: application/pdf; digital
Português
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As the share of wind power in the electricity system rises, the limited predictability of wind power generation becomes increasingly critical for operating a reliable electricity system. In most operational & economic models, the wind power forecast error (WPFE) is often assumed to have a Gaussian or so-called B-distribution. However, these distributions are not suited to fully describe the skewed and heavy-tailed character of WPFE data. In this paper, the Lévy a-stable distribution is proposed as an improved description of the WPFE. Based on 6 years of historical wind power data, three forecast scenarios with forecast horizons ranging from 1 to 24 hours are simulated via a persistence approach. The Lévy a-stable distribution models the WPFE better than the Gaussian or so-called B-distribution, especially for short term forecasts. In a case study, an analysis of historical WPFE data showed improvements over the Gaussian and B-distribution between 137 and 567% in terms of cumulative squared residuals. The method presented allows to quantify the probability of a certain error, given a certain wind power forecast. This new statistical description of the WPFE can hold important information for short term economic & operational (reliability) studies in the field of wind power.

## Exact Multivariate Tests of Asset Pricing Models with Stable Asymmetric Distributions

BEAULIEU, Marie-Claude; DUFOUR, Jean-Marie; KHALAF, Lynda
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 204421 bytes; application/pdf
Português
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In this paper, we propose exact inference procedures for asset pricing models that can be formulated in the framework of a multivariate linear regression (CAPM), allowing for stable error distributions. The normality assumption on the distribution of stock returns is usually rejected in empirical studies, due to excess kurtosis and asymmetry. To model such data, we propose a comprehensive statistical approach which allows for alternative - possibly asymmetric - heavy tailed distributions without the use of large-sample approximations. The methods suggested are based on Monte Carlo test techniques. Goodness-of-fit tests are formally incorporated to ensure that the error distributions considered are empirically sustainable, from which exact confidence sets for the unknown tail area and asymmetry parameters of the stable error distribution are derived. Tests for the efficiency of the market portfolio (zero intercepts) which explicitly allow for the presence of (unknown) nuisance parameter in the stable error distribution are derived. The methods proposed are applied to monthly returns on 12 portfolios of the New York Stock Exchange over the period 1926-1995 (5 year subperiods). We find that stable possibly skewed distributions provide statistically significant improvement in goodness-of-fit and lead to fewer rejections of the efficiency hypothesis.

## Ruin probabilities in tough times - Part 1 - Heavy-traffic approximation for fractionally integrated random walks in the domain of attraction of a nonGaussian stable distribution

Barbe, Ph.; McCormick, W. P.
Tipo: Artigo de Revista Científica
Publicado em 23/01/2011 Português
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Motivated by applications to insurance mathematics, we prove some heavy-traffic limit theorems for process which encompass the fractionally integrated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution.; Comment: 52 pages

## Fractional Laplacian, Levy stable distribution, and time-space models for linear and nonlinear frequency-dependent lossy media

Chen, W.; Holm, S.
Tipo: Artigo de Revista Científica
Publicado em 30/12/2002 Português
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The frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency-squared dependent attenuations. The otherwise non-zeor and non-square frequency dependency occuring in many cases of practical interest is thus often called the anomalous attenuation. In this study, we developed a linear integro-differential equation wave model for the anomalous attenuation by using the space fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers, KZK, and Westervelt equations. A new definition of the fractional Laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hyper-singularity in the conventional fractional Laplacian. Under the Szabo's smallness approximation where attenuation is assumed to be much smaller than the wave number, our linear model is found consistent with arbitrary frequency dependencies. According to the fact that the physical attenuation can be understood a statistic process, the empirical range [0,2] of the power law exponent is explained via the Levy stable distribution theory. It is noted that the power law attentuation underlies fractal microstructures of anomalously attenuative media.

## Ruin probabilities in tough times - Part 2 - Heavy-traffic approximation for fractionally differentiated random walks in the domain of attraction of a nonGaussian stable distribution

Barbe, Ph.; McCormick, W. P.
Tipo: Artigo de Revista Científica
Publicado em 18/02/2011 Português
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Motivated by applications to insurance mathematics, we prove some heavy-traffic limit theorems for processes which encompass the fractionally differentiated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution.; Comment: 17 pages

## Levy stable distribution and [0,2] power law dependence of acoustic absorption on frequency

Chen, W
Tipo: Artigo de Revista Científica
Publicado em 08/05/2005 Português
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The absorption of acoustic wave propagation in a broad variety of lossy media is characterized by an empirical power law function of frequency, w^y. It has long been noted that exponent y ranges from 0 to 2 for diverse media. Recently, the present author developed a fractional Laplacian wave equation to accurately model the power law dissipation, which can be further reduced to the fractional Laplacian diffusion equation. The latter is known underlying the Levy stable distribution theory. Consequently, the parameters y is found to be the Levy stability index, which is known bounded within 0

## Mixed Tempered Stable distribution

Rroji, Edit; Mercuri, Lorenzo
Tipo: Artigo de Revista Científica
Publicado em 29/05/2014 Português
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In this paper we introduce a new parametric distribution, the Mixed Tempered Stable. It has the same structure of the Normal Variance Mean Mixtures but the normality assumption leaves place to a semi-heavy tailed distribution. We show that, by choosing appropriately the parameters of the distribution and under the concrete specification of the mixing random variable, it is possible to obtain some well-known distributions as special cases. We employ the Mixed Tempered Stable distribution which has many attractive features for modeling univariate returns. Our results suggest that it is enough flexible to accomodate different density shapes. Furthermore, the analysis applied to statistical time series shows that our approach provides a better fit than competing distributions that are common in the practice of finance.

## Invariance principles for some FARIMA and nonstationary linear processes in the domain of a stable distribution

Barbe, Ph.; McCormick, W. P.
Tipo: Artigo de Revista Científica
Publicado em 04/07/2010 Português
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We prove some invariance principles for processes which generalize FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution. The limiting processes are extensions of the fractional L\'evy processes. The technique used is interesting in itself; it extends an older idea of splitting a sample into a central part and an extreme one, analyzing each part with different techniques, and then combining the results. This technique seems to have the potential to be useful in other problems in the domain of nonGaussian stable distributions.; Comment: 77 pages, 1 figure

## Regression with an infinite number of observations applied to estimating the parameters of the stable distribution using the empirical characteristic function

van Zyl, J. Martin
Tipo: Artigo de Revista Científica
Publicado em 05/09/2013 Português
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A function of the empirical characteristic function,exists for the stable distribution, which leads to a linear regression and can be used to estimate the parameters. Two approaches are often used, one to find optimal values of t, but these points are dependent on the unknown parameters. And using a fixed number of values for t. In this work the results when all points in an interval is used, thus where least squares using an infinite number of observations,is approximated. It was found that this procedure performs good in small samples.

## On alpha stable distribution of wind driven water surface wave slope

Joelson, Maminirina; Neel, Marie Christine
Tipo: Artigo de Revista Científica
Publicado em 02/10/2008 Português
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We propose a new formulation of the probability distribution function of wind driven water surface slope with an $\alpha$-stable distribution probability. The mathematical formulation of the probability distribution function is given under an integral formulation. Application to represent the probability of time slope data from laboratory experiments is carried out with satisfactory results. We compare also the $\alpha$-stable model of the water surface slopes with the Gram-Charlier development and the non-Gaussian model of Liu et al\cite{Liu}. Discussions and conclusions are conducted on the basis of the data fit results and the model analysis comparison.; Comment: final version of the manuscript: 25 pages

## Estimation of stable distribution parameters from a dependent sample

Tipo: Artigo de Revista Científica
Publicado em 02/05/2014 Português
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Existing methods for the estimation of stable distribution parameters, such as those based on sample quantiles, sample characteristic functions or maximum likelihood generally assume an independent sample. Little attention has been paid to estimation from a dependent sample. In this paper, a method for the estimation of stable distribution parameters from a dependent sample is proposed based on the sample quantiles. The estimates are shown to be asymptotically normal. The asymptotic variance is calculated for stable moving average processes. Simulations from stable moving average processes are used to demonstrate these estimators.; Comment: 18 pages, 1 figure. Most of this paper is included in chapter 3 of my PhD thesis, which is yet to be submitted

## Reward-risk momentum strategies using classical tempered stable distribution

Choi, Jaehyung; Kim, Young Shin; Mitov, Ivan
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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We implement momentum strategies using reward-risk measures as ranking criteria based on classical tempered stable distribution. Performances and risk characteristics for the alternative portfolios are obtained in various asset classes and markets. The reward-risk momentum strategies with lower volatility levels outperform the traditional momentum strategy regardless of asset class and market. Additionally, the alternative portfolios are not only less riskier in risk measures such as VaR, CVaR and maximum drawdown but also characterized by thinner downside tails. Similar patterns in performance and risk profile are also found at the level of each ranking basket in the reward-risk portfolios. Higher factor-neutral returns achieved by the reward-risk momentum strategies are statistically significant and large portions of the performances are not explained by the Carhart four-factor model.; Comment: 38 pages, 6 subfigures, Published version

## Features modeling with an $\alpha$-stable distribution: Application to pattern recognition based on continuous belief functions

Fiche, Anthony; Cexus, Jean-Christophe; Martin, Arnaud; Khenchaf, Ali
Tipo: Artigo de Revista Científica
Publicado em 22/01/2015 Português
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The aim of this paper is to show the interest in fitting features with an $\alpha$-stable distribution to classify imperfect data. The supervised pattern recognition is thus based on the theory of continuous belief functions, which is a way to consider imprecision and uncertainty of data. The distributions of features are supposed to be unimodal and estimated by a single Gaussian and $\alpha$-stable model. Experimental results are first obtained from synthetic data by combining two features of one dimension and by considering a vector of two features. Mass functions are calculated from plausibility functions by using the generalized Bayes theorem. The same study is applied to the automatic classification of three types of sea floor (rock, silt and sand) with features acquired by a mono-beam echo-sounder. We evaluate the quality of the $\alpha$-stable model and the Gaussian model by analyzing qualitative results, using a Kolmogorov-Smirnov test (K-S test), and quantitative results with classification rates. The performances of the belief classifier are compared with a Bayesian approach.

## Elliptical Tempered Stable Distribution and Fractional Calculus

Fallahgoul, Hassan A.; Kim, Young S.
Tipo: Artigo de Revista Científica
Português
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A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, with fractional calculus. Finally, some analytical approximations for the probability density function of tempered infinite divisible distribution, which elliptical tempered stable distributions are a subclass of them, are considered.; Comment: 16 pages, working paper

## Applying least absolute deviation regression to regression-type estimation of the index of a stable distribution using the characteristic function

van Zyl, J. Martin
Tipo: Artigo de Revista Científica
Publicado em 31/07/2013 Português
Relevância na Pesquisa
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Least absolute deviation regression is applied using a fixed number of points for all values of the index to estimate the index and scale parameter of the stable distribution using regression methods based on the empirical characteristic function. The recognized fixed number of points estimation procedure uses ten points in the interval zero to one, and least squares estimation. It is shown that using the more robust least absolute regression based on iteratively re-weighted least squares outperforms the least squares procedure with respect to bias and also mean square error in smaller samples.

## Fully Bayesian Inference for ?-Stable Distributions Using a Poisson Series Representation

Lemke, Tatjana; Riabiz, Marina; Godsill, Simon J.
Fonte: Elsevier Publicador: Elsevier
Tipo: Article; accepted version
Português
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This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.dsp.2015.08.018; In this paper we develop an approach to Bayesian Monte Carlo inference for skewed ?-stable distributions. Based on a series representation of the stable law in terms of infinite summations of random Poisson process arrival times, our framework leads to a simple representation in terms of conditionally Gaussian distributions for certain latent variables. Inference can therefore be carried out straightforwardly using techniques such as auxiliary variables versions of Markov chain Monte Carlo (MCMC) methods. The Poisson series representation (PSR) is further extended to practical application by introducing an approximation of the series residual terms based on exact moment calculations. Simulations illustrate the proposed framework applied to skewed ?-stable simulated and real-world data, successfully estimating the distribution parameter values and being consistent with other (non-Bayesian) approaches. The methods are highly suitable for incorporation into hierarchical Bayesian models, and in this case the conditionally Gaussian structure of our model will lead to very efficient computations compared to other approaches.; Godsill acknowledges partial funding for the work from the EPSRC BTaRoT project EP/K020153/1...

## A proposed method for design of test cases for economic analysis in power systems

Marmolejo-Saucedo,J.A.; Rodríguez-Aguilar,R.
Fonte: UNAM, Centro de Ciencias Aplicadas y Desarrollo Tecnológico Publicador: UNAM, Centro de Ciencias Aplicadas y Desarrollo Tecnológico
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2015 Português
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Nowadays, in power systems, we still lack the existence of standardized test systems that can be used to benchmark the performance and solution quality of proposed optimization techniques. Several authors report that the electric load pattern is very complex. It is therefore necessary to develop new methods for design of test cases for economic analysis in power systems. Therefore, we compared two methods to generate test systems: time series model and a method simulating stable random variables based on the use of Chambers-Mallows-Stuck. This paper describes a method for simulating stable random variables in the generation of test systems for economic analysis in power systems. A study focused on generating test electrical systems through stable distribution to model for unit commitment problem in electrical power systems. Usually, the instances of test systems in unit commitment are generated using normal distribution, but the behavior of electrical demand does not follow a normal distribution; in this work, simulation data are based on a new method. For empirical analysis, we used three original systems to obtain the demand behavior and thermal production costs. Numerical results illustrate the applicability of the proposed method by solving several unit commitment problems directly and through the Lagrangian relaxation of the original problem.

## Asymptotic behavior of the daily increment distribution of the IPC, the mexican stock market index

Coronel-Brizio,H.F.; Hernandez-Montoya,A.R.
Fonte: Sociedad Mexicana de Física Publicador: Sociedad Mexicana de Física
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2005 Português
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In this work, a statistical analysis of the distribution of daily fluctuations of the IPC, the Mexican Stock Market Index is presented. A sample of the IPC covering the 13-year period 04/19/1990 - 08/21/2003 was analyzed and the cumulative probability distribution of its daily logarithmic variations studied. Results show that the cumulative distribution function for extreme variations, can be described by a Pareto-Levy model with shape parameters α= 3.634 ± 0.272 and α= 3.540 ± 0.278 for its positive and negative tails, respectively. This result is consistent with previous studies, where it has been found that 2.5 < α < 4 for other financial markets worldwide.