Calibrated FFT-based density approximations for -stable distributions

Menn, Christian; Rachev, Svetlozar T.

doi:10.1016/j.csda.2005.03.004

Cited by 44 publications

(47 citation statements)

References 7 publications

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“…On the meanwhile its characteristic function is explicitly writable. This is the same case as approximating the α-stable distribution in Menn and Rachev (2004), by which the Fourier transformation is used to approximate the density of the variable based on its characteristic function. This motivates us to use the technique to approximate the density of the return in the GHICA procedure.…”

Section: Normal Inverse Gaussian (Nig) Distribution and Fast Fourier mentioning

confidence: 99%

GHICA — Risk analysis with GH distributions and independent components

Chen

Härdle

Spokoiny

2010

Journal of Empirical Finance

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Over recent years, study on risk management has been prompted by the Basel committee for regular banking supervisory. There are however limitations of some widely-used risk management methods that either calculate risk measures under the Gaussian distributional assumption or involve numerical difficulty. The primary aim of this paper is to present a realistic and fast method, GHICA, which overcomes the limitations in multivariate risk analysis. The idea is to first retrieve independent components (ICs) out of the observed high-dimensional time series and then individually and adaptively fit the resulting ICs in the generalized hyperbolic (GH) distributional framework. For the volatility estimation of each IC, the local exponential smoothing technique is used to achieve the best possible accuracy of estimation. Finally, the fast Fourier transformation technique is used to approximate the density of the portfolio returns.The proposed GHICA method is applicable to covariance estimation as well. It is compared with the dynamic conditional correlation (DCC) method based on the simulated data with d = 50 GH distributed components. We further implement the GHICA method to calculate risk measures given 20-dimensional German DAX portfolios and a dynamic exchange rate portfolio. Several alternative methods are considered as well to compare the accuracy of calculation with the GHICA one.

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Section: Normal Inverse Gaussian (Nig) Distribution and Fast Fourier mentioning

confidence: 99%

GHICA — Risk analysis with GH distributions and independent components

Chen

Härdle

Spokoiny

2010

Journal of Empirical Finance

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“…Experimentally, the committed absolute error is in the order of 10 −5 , but relative error goes as high as 10 −2 . Menn and Rachev (2006) propose a method based on a refinement of the FFT to increase precision in the central part of the PDF. The tails of the distribution are calculated via the Bergström asymptotic series expansion (Zolotarev 1986), which provides an alternative expression as an infinite sum of decaying terms.…”

Section: Numerical Computation Of α-Stable Distributionsmentioning

confidence: 99%

“…Besides, the non-existence of moments of order two or higher (except in the Gaussian case) increases the difficulty in estimating their parameters to fit real data. Several authors have addressed both the numerical evaluation of the PDF or CDF of α-stable distributions (Nolan 1997;Mittnik, Doganoglu, and Chenyao 1999a;Belov 2005;Menn and Rachev 2006;Górska and Penson 2011) and the estimation of their parameters (Fama and Roll 1971;Koutrouvelis 1981;McCulloch 1986;Mittnik, Rachev, Doganoglu, and Chenyao 1999b;Nolan 2001;Fan 2006;Salas-Gonzalez, Kuruoglu, and Ruiz 2009) and have proposed different methods and algorithms for these purposes (details are discussed in Section 2). Some authors also provide implementations of their own or relying on others' methods, offering software solutions or packages in various common computer languages.…”

Section: Introductionmentioning

confidence: 99%

“…Some authors also provide implementations of their own or relying on others' methods, offering software solutions or packages in various common computer languages. However, some of these solutions are not fully operational in their public domain versions and do not take advantage of multi-core processors (Nolan 2006), have limited accuracy (Belov 2005;Menn and Rachev 2006) or take a computation time that makes results unaffordable for several practical applications (Veillete 2010;Liang and Chen 2013).…”

Section: Introductionmentioning

confidence: 99%

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libstable: Fast, Parallel, and High-Precision Computation of α-Stable Distributions in R, C/C++, and MATLAB

Royuela-del-Val¹,

Simmross-Wattenberg²,

Alberola-López³

2017

J. Stat. Soft.

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Abstractα-stable distributions are a family of well-known probability distributions. However, the lack of closed analytical expressions hinders their application. Currently, several tools have been developed to numerically evaluate their density and distribution functions or to estimate their parameters, but available solutions either do not reach sufficient precision on their evaluations or are excessively slow for practical purposes. Moreover, they do not take full advantage of the parallel processing capabilities of current multi-core machines. Other solutions work only on a subset of the α-stable parameter space. In this paper we present an R package and a C/C++ library with a MATLAB front-end that permit parallelized, fast and high precision evaluation of density, distribution and quantile functions, as well as random variable generation and parameter estimation of α-stable distributions in their whole parameter space. The described library can be easily integrated into third party developments.

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“…Further refinements to this FFT-based approximation of stable densities 253 were described by [33], which included numerically calculating the integral in 254 (12) using Simpson's rule, and replacing the linear interpolation with cubic 255 splines. In practice, we found that there was no noticeable advantage to 256 using these more costly techniques in the long memory context.…”

mentioning

confidence: 99%

Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models

Graves

Franzke²,

Watkins

et al. 2017

Physica A: Statistical Mechanics and its Applications

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Please cite this article as: T. Graves, et al., Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.01.028. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights• Self-Similarity has two contributions: Long-range dependence and heavy-tailed jumps• Systematic simultaneous estimation of long-range dependence and heavy-tail distribution parameters• Development of a novel Bayesian method for estimation of these two parameters• Method flexible to allow choice of heavy-tailed distribution (e.g. t-or α-stable distributed)• Successful demonstration of effectiveness and accuracy on synthetic data 1 AbstractLong-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 selfsimilarity 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 * Corresponding author Email addresses: christian.franzke@uni-hamburg.de (Christian L. E. Franzke)Preprint submitted to Physica A December 14, 2016LRD and heavy-tailed distribution parameters.

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Calibrated FFT-based density approximations for -stable distributions

Cited by 44 publications

References 7 publications

GHICA — Risk analysis with GH distributions and independent components

GHICA — Risk analysis with GH distributions and independent components

libstable: Fast, Parallel, and High-Precision Computation of α-Stable Distributions in R, C/C++, and MATLAB

Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models

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