Heavy-tailed Distributions (heavy-tailed + distribution)

Distribution by Scientific Domains

Selected Abstracts

Portfolio Value-at-Risk with Heavy-Tailed Risk Factors

Paul Glasserman
This paper develops efficient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR. [source]

Analysis of historical landslide time series in the Emilia-Romagna region, northern Italy

Mauro Rossi
Abstract A catalogue of historical landslides, 1951,2002, for three provinces in the Emilia-Romagna region of northern Italy is presented and its statistical properties studied. The catalogue consists of 2255 reported landslides and is based on historical archives and chronicles. We use two measures for the intensity of landsliding over time: (i) the number of reported landslides in a day (DL) and (ii) the number of reported landslides in an event (Sevent), where an event is one or more consecutive days with landsliding. From 1951,2002 in our study area there were 1057 days with 1 , DL ,?45 landslides per day, and 596 events with 1 , Sevent , 129 landslides per event. In the first set of analyses, we find that the probability density of landslide intensities in the time series are power-law distributed over at least two-orders of magnitude, with exponent of about ,20. Although our data is a proxy for landsliding built from newspaper reports, it is the first tentative evidence that the frequency-size of triggered landslide events over time (not just the landslides in a given triggered event), like earthquakes, scale as a power-law or other heavy-tailed distributions. If confirmed, this could have important implications for risk assessment and erosion modelling in a given area. In our second set of analyses, we find that for short antecedent rainfall periods, the minimum amount of rainfall necessary to trigger landslides varies considerably with the intensity of the landsliding (DL and Sevent); whereas for long antecedent periods the magnitude is largely independent of the cumulative amount of rainfall, and the largest values of landslide intensity are always preceded by abundant rainfall. Further, the analysis of the rainfall trend suggests that the trigger of landslides in the study area is related to seasonal rainfall. Copyright 2010 John Wiley & Sons, Ltd. [source]

Power laws without complexity

Andrew R. Solow
Abstract Power laws have been invoked in describing the sizes of a wide variety of objects in evolution and ecology. The apparent ubiquity of power laws is commonly attributed to a form of complex behaviour called self-organizing criticality. It is shown that power law behaviour inevitably arises from the statistics of large values from heavy-tailed distributions and that such distributions can be generated by processes that do not involve self-organized criticality. It follows that power law behaviour cannot be taken as prima facie evidence of self-organizing criticality. [source]

Double hierarchical generalized linear models (with discussion)

Youngjo Lee
Summary., We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h -likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities. [source]

Asymptotic self-similarity and wavelet estimation for long-range dependent fractional autoregressive integrated moving average time series with stable innovations

Stilian Stoev
Primary 60G18; 60E07; Secondary 62M10; 63G20 Abstract., Methods for parameter estimation in the presence of long-range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long-range dependence in the case of heavy-tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/, for long-range dependent FARIMA time series with symmetric , -stable (1 < , < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self-similar with parameter H. When the parameter , is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H , 1/,. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet-based estimation method used here is semi-parametric, it allows for a more robust treatment of long-range dependent data than parametric methods. [source]

Subexponential Distributions , Large Deviations with Applications to Insurance and Queueing Models

Aleksandras Baltr
Summary This paper presents a fine large-deviations theory for heavy-tailed distributions whose tails are heavier than exp(,,t and have finite second moment. Asymptotics for first passage times are derived. The results are applied to estimate the finite time ruin probabilities in insurance as well as the busy period in a GI/G/1 queueing model. [source]