Heavy-tailed Marginal Distributions Research Articles

Causality in extremes of time series

Consider two stationary time series with heavy-tailed marginal distributions. We aim to detect whether they have a causal relation, that is, if a change in one causes a change in the other. Usual methods for causal discovery are not well suited if the causal mechanisms only appear during extreme events. We propose a framework to detect a causal structure from the extremes of time series, providing a new tool to extract causal information from extreme events. We introduce the causal tail coefficient for time series, which can identify asymmetrical causal relations between extreme events under certain assumptions. This method can handle nonlinear relations and latent variables. Moreover, we mention how our method can help estimate a typical time difference between extreme events. Our methodology is especially well suited for large sample sizes, and we show the performance on the simulations. Finally, we apply our method to real-world space-weather and hydro-meteorological datasets.

Joint inference on extreme expectiles for multivariate heavy-tailed distributions

Expectiles induce a law-invariant, coherent and elicitable risk measure that has received substantial attention in actuarial and financial risk management contexts. A number of recent papers have focused on the behaviour and estimation of extreme expectile-based risk measures and their potential for risk assessment was highlighted in financial and actuarial real data applications. Joint inference of several extreme expectiles has however been left untouched; in fact, even the inference about a marginal extreme expectile turns out to be a difficult problem in finite samples, even though an accurate idea of estimation uncertainty is crucial for the construction of confidence intervals in applications to risk management. We investigate the joint estimation of extreme marginal expectiles of a random vector with heavy-tailed marginal distributions, in a general extremal dependence model. We use these results to derive corrected confidence regions for extreme expectiles, as well as a test for the equality of tail expectiles. The methods are showcased in a finite-sample simulation study and on real financial data.

Conditional marginal expected shortfall

In the context of bivariate random variables $\left (Y^{(1)},Y^{(2)}\right )$ , the marginal expected shortfall, defined as $\mathbb {E}\left (Y^{(1)}|Y^{(2)} \ge Q_{2}(1-p)\right )$ for p small, where Q2 denotes the quantile function of Y(2), is an important risk measure, which finds applications in areas like, e.g., finance and environmental science. Our paper pioneers the statistical modeling of this risk measure when the random variables of main interest $\left (Y^{(1)},Y^{(2)}\right )$ are observed together with a random covariate X, leading to the concept of the conditional marginal expected shortfall. The asymptotic behavior of an estimator for this conditional marginal expected shortfall is studied for a wide class of conditional bivariate distributions, with heavy-tailed marginal conditional distributions, and where p tends to zero at an intermediate rate. The finite sample performance is evaluated on a small simulation experiment. The practical applicability of the proposed estimator is illustrated on flood claim data.

Intermittency and infinite variance: the case of integrated supOU processes

SupOU processes are superpositions of Ornstein-Uhlenbeck type processes with a random intensity parameter. They are stationary processes whose marginal distribution and dependence structure can be specified independently. Integrated supOU processes have then stationary increments and satisfy central and non-central limit theorems. Their moments, however, can display an unusual behavior known as “intermittency”. We show here that intermittency can also appear when the processes have a heavy tailed marginal distribution and, in particular, an infinite variance.

Testing the existence of moments for GARCH processes

It is generally admitted that many financial time series have heavy tailed marginal distributions. When time series models are fitted on such data, the non-existence of appropriate moments may invalidate standard statistical tools used for inference. Moreover, the existence of moments can be crucial for risk management, for instance when risk is measured through the expected shortfall. This paper considers testing the existence of moments in the framework of GARCH processes. While the second-order stationarity condition does not depend on the distribution of the innovation, higher-order moment conditions involve moments of the independent innovation process. We propose tests for the existence of high moments of the returns process which are based on the joint asymptotic distribution of the Quasi-Maximum Likelihood (QML) estimator of the volatility parameters and empirical moments of the residuals. A bootstrap procedure is proposed to improve the finite-sample performance of our test. To achieve efficiency gains we consider non Gaussian QML estimators founded on reparameterizations of the GARCH model, and we discuss optimality issues. Monte Carlo experiments and an empirical study illustrate the asymptotic results.

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets.

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.

Heavy-tailed fractional Pearson diffusions

We define heavy-tailed fractional reciprocal gamma and Fisher–Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher–Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher–Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation.

Additive watermark detection in the wavelet domain using 2D-GARCH model

Watermark detection can be generally achieved by statistical decision methods. So, selection of an accurate statistical model is one of the major issues in watermark detection. Usually, watermark detectors assume the wavelet coefficients to be independent and identically distributed, therefore, they cannot take into account important characteristics of the wavelet coefficients such as dependency and heteroscedasticity. This paper presents a novel detector for wavelet domain additive image watermarking. We use two dimensional generalized autoregressive conditional heteroscedasticity (2D-GARCH) model for the wavelet coefficients. This model can capture the dependency, heteroscedasticity and heavy-tailed marginal distribution of these coefficients. Based on 2D-GARCH model, we design a new watermark detector and derive its receiver operating characteristics. Experimental results demonstrate the efficiency of proposed detector and its higher performance compared to alternative watermarking methods in the literature.

A unified approach to self-normalized block sampling

The inference procedure for the mean of a stationary time series is usually quite different under various model assumptions because the partial sum process behaves differently depending on whether the time series is short or long-range dependent, or whether it has a light or heavy-tailed marginal distribution. In the current paper, we develop an asymptotic theory for the self-normalized block sampling, and prove that the corresponding block sampling method can provide a unified inference approach for the aforementioned different situations in the sense that it does not require the a priori estimation of auxiliary parameters. Monte Carlo simulations are presented to illustrate its finite-sample performance. The R function implementing the method is available from the authors.

Uncertainty evaluation of copula analysis of hydrological droughts in the East River basin, China

The joint probability behaviors of extreme hydro-meteorological events, such as droughts, have been receiving increasing attentions in recent years. Since extreme hydro-meteorological events are reflected by more than one variable, such as duration and intensity, copula functions have been widely applied. However, uncertainties of copula-based analysis of hydrological droughts as a result of selection of marginal distribution and copulas have not yet received significant concerns. The aim of this study is to evaluate such uncertainties based on Bayesian approach. The method is used to analyze hydrological drought in the East River basin (China), which is the principal supplier of water resources for megacities in the Pearl River Delta and also for Hong Kong. The results indicate that the credible intervals of most likely design drought events with a return period of 20years in terms of drought severity and duration are considerably large at all stations in the East River basin, covering the drought event curves at return periods of 10 and 50years. Also the influences of heavy-tailed marginal distribution on the uncertainty evaluation of joint distribution have been investigated. Results show that the stronger the heavy-tailed marginal distribution, the greater the uncertainty of the joint distribution, especially for the extreme event. Results of this study provide a technical reference for uncertainty evaluation in copula-based analysis of drought events at regional and global scales. The large credible interval of drought severity and duration greatly challenges measures of mitigation of drought hazards and water resource management.

Statistical modeling and denoising Wigner–Ville distribution

Studying the properties of the Wigner-Ville distribution (wvd) and its smoothed versions such as smoothed pseudo-WVD (spwvd), we demonstrate that they have significantly non-Gaussian statistics. Also, we investigate the presence of two-dimensional heteroscedasticity in them for different signals based on employing Lagrange multiplier (LM) procedure. Therefore, we employ a heteroscedastic model called two-dimensional generalized autoregressive conditional heteroscedastic (2-D garch) for statistical modeling of these distributions. This modeling captures the characteristics of WVD and SPWVD, such as heavy tailed marginal distribution, and the dependencies among them. Since the performance of WVD and its smoothed versions degrade in the presence of additive noise, we design a novel Bayesian estimator for estimating the clean distributions based on garch modeling. Also, estimating the instantaneous frequency (if) curves of signals in presence of noise based on WVD and its smoothed versions is an interesting topic in the radar domain. So, we apply the denoised distributions for estimating the if. Experimental results demonstrate the efficiency of proposed method in denoising wvd and SPWVD and also performance improvement for if estimation in utilizing the denoised distributions.

Wavelet domain Bayesian processor for speckle removal in medical ultrasound images

Speckle reduction is a prerequisite for many ultrasound image processing tasks. In this study, the authors introduce a novel speckle suppression method for ultrasound images, based on statistical modelling of wavelet coefficients. First, the authors demonstrate that the wavelet coefficients of log-transformed ultrasound images have significantly non-Gaussian statistics and a two-dimensional heteroscedasticity exists in them. Previously, they described the overall multiscale wavelet transform of the log-transformed ultrasound images using two-dimensional generalised autoregressive conditional heteroscedastic (2D GARCH) model. In this study, they introduce a new heteroscedastic model, that is, 2D-GARCH generalised Gaussian (2D-GARCH-GG) as an extension of 2D-GARCH model. This new model can capture heavy-tailed marginal distribution and the intrascale dependencies of wavelet coefficients. Also, 2D-GARCH-GG model introduces additional flexibility in the model formulation in comparison with 2D-GARCH model, which results in better characterisation of ultrasound images subbands and improved restoration in noisy environments. In consequence, the authors introduce maximum a-posteriori estimator, based on GARCH-GG modelling to estimate the clean wavelet coefficients. In order to evaluate the performance of the proposed method in speckle suppression, they compare it with other denoising methods applied on some artificially speckled and actual ultrasound images and we verify the performance improvement in utilising the new strategies.

Modeling Functional Data with Spatially Heterogeneous Shape Characteristics

We propose a novel class of models for functional data exhibiting skewness or other shape characteristics that vary with spatial or temporal location. We use copulas so that the marginal distributions and the dependence structure can be modeled independently. Dependence is modeled with a Gaussian or t-copula, so that there is an underlying latent Gaussian process. We model the marginal distributions using the skew t family. The mean, variance, and shape parameters are modeled nonparametrically as functions of location. A computationally tractable inferential framework for estimating heterogeneous asymmetric or heavy-tailed marginal distributions is introduced. This framework provides a new set of tools for increasingly complex data collected in medical and public health studies. Our methods were motivated by and are illustrated with a state-of-the-art study of neuronal tracts in multiple sclerosis patients and healthy controls. Using the tools we have developed, we were able to find those locations along the tract most affected by the disease. However, our methods are general and highly relevant to many functional data sets. In addition to the application to one-dimensional tract profiles illustrated here, higher-dimensional extensions of the methodology could have direct applications to other biological data including functional and structural magnetic resonance imaging (MRI).

Two Novel Bayesian Multiscale Approaches for Speckle Suppression in SAR Images

Speckle suppression is a prerequisite for many synthetic aperture radar (SAR) image-processing tasks. Previously, we introduced a Bayesian-based speckle-suppression method that employed the 2-D generalized autoregressive conditional heteroscedasticity (2D-GARCH) model for wavelet coefficients of log-transformed SAR images. Based on this method, we propose two new Bayesian speckle-suppression approaches in this paper. In the first approach, we introduce a new heteroscedastic model, i.e., the 2D-GARCH Mixture (2D-GARCH-M) model, as an extension of the 2D-GARCH model. This new model can capture the characteristics of wavelet coefficients. Also, the 2D-GARCH-M model introduces additional flexibility in the model formulation in comparison with the 2D-GARCH model, which results in better characterization of SAR image subbands and improved restoration in noisy environments. Then, we design a Bayesian estimator for estimating the clean-image wavelet coefficients based on 2D-GARCH-M modeling. In the second approach, the logarithm of an image is analyzed by means of the curvelet transform instead of wavelet transform. Then, we study the statistical properties of curvelet coefficients, and we demonstrate that the 2D-GARCH model can capture the characteristics of curvelet coefficients, such as heavy tailed marginal distribution, and the dependences among them. Consequently, under the 2D-GARCH model, we design a Bayesian estimator for estimating the clean-image curvelet coefficients. Finally, we compare these methods with other denoising methods applied on artificially speckled and actual SAR images, and we verify the performance improvement in utilizing the new strategies.

Speckle Suppression in SAR Images Using the 2-D GARCH Model

A novel Bayesian-based speckle suppression method for Synthetic Aperture Radar ( SAR) images is presented that preserves the structural features and textural information of the scene. First, the logarithmic transform of the original image is analyzed into the multiscale wavelet domain. We show that the wavelet coefficients of SAR images have significantly non-Gaussian statistics that are best described by the 2-D GARCH model. By using the 2-D GARCH model on the wavelet coefficients, we are capable of taking into account important characteristics of wavelet coefficients, such as heavy tailed marginal distribution and the dependencies between the coefficients. Furthermore, we use a maximum a posteriori (MAP) estimator for estimating the clean image wavelet coefficients. Finally, we compare our proposed method with various speckle suppression methods applied on synthetic and actual SAR images and we verify the performance improvement in utilizing the new strategy.

Numerical simulation for solute transport in fractal porous media

A modified Fokker-Planck equation with continuous source for solute transport in fractal porous media is considered. The dispersion term of the governing equation uses a fractional-order derivative and the diffusion coefficient can be time and scale dependent. In this paper, numerical solution of the modified Fokker-Planck equation is proposed. The effects of different fractional orders and fractional power functions of time and distance are numerically investigated. The results show that motions with a heavy tailed marginal distribution can be modelled by equations that use fractional-order derivatives and/or time and scale dependent dispersivity.

A TWO - STEP MAXIMUM - LIKELIHOOD IDENTIFICATION OF NON - GAUSSIAN SYSTEMS

ARMA modeling of many economic time series leads to processes with heavy-tailed marginal distribution. We present methods of estimating the parameters of such processes. Asymptotic properties of the full information maximum likelihood and partially adaptive estimates are discussed. We give an asymptotic description of the estimation error process in both cases. The results are generalizations of (Philips, 1994) and (Gerencsér, 1990).

Point process convergence of stochastic volatility processes with application to sample autocorrelation

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.

Point process convergence of stochastic volatility processes with application to sample autocorrelation

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.