Class Of Heavy-tailed Distributions Research Articles

Minimum of heavy-tailed random variables is not heavy tailed

<abstract><p>By constructing an appropriate example, we show that the class of heavy-tailed distributions is not closed under minimum. We provide two independent heavy-tailed random variables, such that their minimum is not heavy tailed. In addition, we establish a few properties of the distributions considered in the example.</p></abstract>

MCMC Algorithms for Posteriors on Matrix Spaces

We study Markov chain Monte Carlo (MCMC) algorithms for target distributions defined on matrix spaces. Such an important sampling problem has yet to be analytically explored. We carry out a major step in covering this gap by developing the proper theoretical framework that allows for the identification of ergodicity properties of typical MCMC algorithms, relevant in such a context. Beyond the standard Random-Walk Metropolis (RWM) and preconditioned Crank–Nicolson (pCN), a contribution of this article in the development of a novel algorithm, termed the “Mixed” pCN (MpCN). RWM and pCN are shown not to be geometrically ergodic for an important class of matrix distributions with heavy tails. In contrast, MpCN is robust across targets with different tail behavior and has very good empirical performance within the class of heavy-tailed distributions. Geometric ergodicity for MpCN is not fully proven in this work, as some remaining drift conditions are quite challenging to obtain owing to the complexity of the state space. We do, however, make a lot of progress toward a proof, and show in detail the last steps left for future work. We illustrate the computational performance of the various algorithms through numerical applications, including calibration on real data of a challenging model arising in financial statistics. Supplementary materials for this article are available online.

ALTERNATIVE ROBUST ESTIMATORS FOR PARAMETERS OF THE LINEAR REGRESSION MODEL

This paper considers parameter estimation of the linear regression model with Ramsay-Novick (RN) distributed errors, focusing on its use as an aid to robustness. Positioning within the class of heavy tailed distributions, RN distribution can be defined as the modification of unbounded influence function of a non-robust density so that it has more resistance to outliers. Potential use of this robust density have so far been assessed in Bayesian settings on real data examples and there is a lack of performance assessment for finite samples in classical approach. This study therefore explores its robustness properties when used as error distribution in comparison to normal as well as other alternating heavy-tailed distributions like Laplace and Student-t. An extensive simulation study was conducted for this purpose under different settings of sample size, model parameters and outlier percantages. An efficient data generation of this distribution through random-walk Metropolis algorithm is here also suggested. The results were supported by a real world application on famously known as Brownlee’s stack loss plant data.

A New Class of Heavy-Tailed Distributions: Modeling and Simulating Actuarial Measures

Statistical distributions play a prominent role for modeling data in applied fields, particularly in actuarial, financial sciences, and risk management fields. Among the statistical distributions, the heavy-tailed distributions have proven the best choice to use for modeling heavy-tailed financial data. The actuaries are often in search of such types of distributions to provide the best description of the actuarial and financial data. This study presents a new power transformation to introduce a new family of heavy-tailed distributions useful for modeling heavy-tailed financial data. A submodel, namely, heavy-tailed beta-power transformed Weibull model is considered to demonstrate the adequacy of the proposed method. Some actuarial measures such as value at risk, tail value at risk, tail variance, and tail variance premium are calculated. A brief simulation study based on these measures is provided. Finally, an application to the insurance loss dataset is analyzed, which revealed that the proposed distribution is a superior model among the competitors and could potentially be very adequate in describing and modeling actuarial and financial data.

A new extended alpha power transformed family of distributions: properties, characterizations and an application to a data set in the insurance sciences

Heavy tailed distributions are useful for modeling actuarial and financial risk management problems. Actuaries often search for finding distributions that provide the best fit to heavy tailed data sets. In the present work, we introduce a new class of heavy tailed distributions of a special sub-model of the proposed family, called a new extended alpha power transformedWeibull distribution, useful for modeling heavy tailed data sets. Mathematical properties along with certain characterizations of the proposed distribution are presented. Maximum likelihood estimates of the model parameters are obtained. A simulation study is provided to evaluate the performance of the maximum likelihood estimators. Actuarial measures such as Value at Risk and Tail Value at Risk are also calculated. Further, a simulation study based on the actuarial measures is done. Finally, an application of the proposed model to a heavy tailed data set is presented. The proposed distribution is compared with some well-known (i) two-parameter models, (ii) three-parameter models and (iii) four-parameter models.

Type-I heavy tailed family with applications in medicine, engineering and insurance.

In the present study, a new class of heavy tailed distributions using the T-X family approach is introduced. The proposed family is called type-I heavy tailed family. A special model of the proposed class, named Type-I Heavy Tailed Weibull (TI-HTW) model is studied in detail. We adopt the approach of maximum likelihood estimation for estimating its parameters, and assess the maximum likelihood performance based on biases and mean squared errors via a Monte Carlo simulation framework. Actuarial quantities such as value at risk and tail value at risk are derived. A simulation study for these actuarial measures is conducted, proving that the proposed TI-HTW is a heavy-tailed model. Finally, we provide a comparative study to illustrate the proposed method by analyzing three real data sets from different disciplines such as reliability engineering, bio-medical and financial sciences. The analytical results of the new TI-HTW model are compared with the Weibull and some other non-nested distributions. The Baysesian analysis is discussed to measure the model complexity based on the deviance information criterion.

Bayesian inference of mixed-effects ordinary differential equations models using heavy-tailed distributions

A mixed-effects ordinary differential equation (ODE) model is proposed to describe complex dynamical systems. In order to make the inference of ODE parameters robust against the outlying observations and subjects, a class of heavy-tailed distributions is applied to model the random effects of ODE parameters and measurement errors in the data. The heavy-tailed distributions are so flexible that they include the conventional normal distribution as a special case. An MCMC method is proposed to make inferences on ODE parameters within a Bayesian hierarchical framework. The proposed method is demonstrated by estimating a pharmacokinetic mixed-effects ODE model. The finite sample performance of the proposed method is evaluated using some simulation studies.

Extreme M-quantiles as risk measures: From $L^{1}$ to $L^{p}$ optimization

The class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of $L^{p}$ optimization. These generalized $L^{p}$-quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for $1<p<2$. In this paper, we investigate their estimation from the perspective of extreme values in the class of heavy-tailed distributions. We construct estimators of the intermediate $L^{p}$-quantiles and establish their asymptotic normality in a dependence framework motivated by financial and actuarial applications, before extrapolating these estimates to the very far tails. We also investigate the potential of extreme $L^{p}$-quantiles as a tool for estimating the usual quantiles and expectiles themselves. We show the usefulness of extreme $L^{p}$-quantiles and elaborate the choice of $p$ through applications to some simulated and financial real data.

Fitting phase\u2013type scale mixtures to heavy\u2013tailed data and distributions

We consider the fitting of heavy tailed data and distributions with a special attention to distributions with a non–standard shape in the “body” of the distribution. To this end we consider a dense class of heavy tailed distributions introduced in Bladt et al. (Scand. Actuar. J., 573–591 2015), employing an EM algorithm for the maximum likelihood estimation of its parameters. We present methods for fitting to observed data, histograms, censored data, as well as to theoretical distributions. Numerical examples are provided with simulated data and a benchmark reinsurance dataset. Empirical examples show that the methods will in most cases adequately fit both body and tail simultaneously.

A New Class of Heavy-Tailed Distribution and the Stock Market Returns in Germany

As the fourth largest economy over the world, Germany’s financial sector plays a key role in the global economy. As one of the most important components of the financial sector, the equity market played a more and more important role. Thus, risk management of its stock market is crucial for welfare of its market participants. To account for the two stylized facts, volatility clustering and conditional heavy tails, we take advantage of the framework in Guo (2016) and consider empirical performance of the GARCH model with normal reciprocal inverse Gaussian distribution in fitting the German stock return series. Our results indicate the NRIG distribution has superior performance in fitting the stock market returns.

GARCH Models with a New Class of Heavy-Tailed Distribution for the Silver Returns

After serving as a medium of exchange for the human society, silver is still widely used in our daily life. From the jewelry, electronic and electrical industries as well as medicine, optics, the power industry, automotive industry and many other industries, silver is still playing a very active role. In addition to the industrial usage, silver also serves as an investment tool for many financial institutions. Thus, it is crucial to develop effective quantitative risk management tool for those financial institutions. In this paper, we investigate the conditional heavy tails of daily silver spot returns under the GARCH framework. Our results indicate that that it is important to introduce heavy-tailed distributions to the GARCH framework and the normal reciprocal inverse Gaussian (NRIG) distribution, a newly-developed distribution, has the best empirical performance in capture the daily silver spot returns dynamics.

On the random max-closure for heavy-tailed random variables

In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X1,X2, . . . , which states that, given distributions \( {F}_{X_1} \),\( {F}_{X_2} \), . . . from some class of heavy-tailed distributions, the distribution of the random maximum X(η) := max{0,X1, . . . , Xη} or random maximum S(η) := max{0, S1, . . . , Sη} belongs to the same class of heavy-tailed distributions. Here, Sn = X1 + · · · + Xn, n ≥ 1, and η is a counting random variable, independent of {X1,X2, . . . }. We provide the conditions for the random max-closure property in the case of classes Open image in new window and Open image in new window.

Convolution and convolution-root properties of long-tailed distributions

We obtain a number of new general properties, related to the closedness of the class of long-tailed distributions under convolutions, that are of interest themselves and may be applied in many models that deal with “plus” and/or “max” operations on heavy-tailed random variables. We analyse the closedness property under convolution roots for these distributions. Namely, we introduce two classes of heavy-tailed distributions that are not long-tailed and study their properties. These examples help to provide further insights and, in particular, to show that the properties to be both long-tailed and so-called “generalised subexponential” are not preserved under the convolution roots. This leads to a negative answer to a conjecture of Embrechts and Goldie (J. Austral. Math. Soc. (Ser. A) 29, 243–256 1980, Stoch. Process. Appl. 13, 263–278 1982) for the class of long-tailed and generalised subexponential distributions. In particular, our examples show that the following is possible: an infinitely divisible distribution belongs to both classes, while its Levy measure is neither long-tailed nor generalised subexponential.

Assessing The Relative Performance Of Heavy-Tailed Distributions: Empirical Evidence From The Johannesburg Stock Exchange

&lt;p&gt;It has been well documented that the empirical distribution of daily logarithmic returns from financial market variables is characterized by excess kurtosis and skewness. In order to capture such properties in financial data, heavy-tailed and asymmetric distributions are required to overcome shortfalls of the widely exhausted classical normality assumption. In the context of financial forecasting and risk management, the accuracy in modeling the underlying returns distribution plays a vital role. For example, risk management tools such as value-at-risk (VaR) are highly dependent on the underlying distributional assumption, with particular focus being placed at the extreme tails. Hence, identifying a distribution that best captures all aspects of the given financial data may provide vast advantages to both investors and risk managers. In this paper, we investigate major financial indices on the Johannesburg Stock Exchange (JSE) and fit their associated returns to classes of heavy tailed distributions. The relative adequacy and goodness-of-fit of these distributions are then assessed through the robustness of their respective VaR estimates. Our results indicate that the best model selection is not only variant across the indices, but also across different VaR levels and the dissimilar tails of return series.&lt;/p&gt;

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.

Calculation of ruin probabilities for a dense class of heavy tailed distributions

In this paper, we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional phase-type distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.

A Marshall–Olkin family of heavy-tailed distributions which includes the lognormal one

A new class of heavy-tailed distribution functions,, containing the lognormal distribution as a particular case is introduced. The class thus obtained depends on a set of three parameters, incorporating an additional distribution to the classical lognormal one. This new class of heavy-tailed distribution is presented as an alternative to other useful heavy-tailed distributions, such as the lognormal, Weibull, and Pareto distributions. The density and distribution functions of this new class are given by a closed expression which allows us to easily compute probabilities, quantiles, moments, and related measurements. Finally, some applications are shown as examples.

Modelling conflicting information using subexponential distributions and related classes

In the Bayesian modelling the data and the prior information concerning a certain parameter of interest may conflict, in the sense that the information carried by them disagree. The most common form of conflict is the presence of outlying information in the data, which may potentially lead to wrong posterior conclusions. To prevent this problem we use robust models which aim to control the influence of the atypical information in the posterior distribution. Roughly speaking, we conveniently use heavy-tailed distributions in the model in order to resolve conflicts in favour of those sources of information which we believe is more credible. The class of heavy-tailed distributions is quite wide and the literature have been concerned in establishing conditions on the data and prior distributions in order to reject the outlying information. In this work we focus on the subexponential and \(\mathfrak L \) classes of heavy-tailed distributions, in which we establish sufficient conditions under which the posterior distribution automatically rejects the conflicting information.

Identifiability of flow distributions from link measurements with applications to computer networks

We study the problem of identifiability of distributions of flows on a graph from aggregate measurements collected on its edges. This is a canonical example of a statistical inverse problem motivated by recent developments in computer networks. In this paper (i) we introduce a number of models for multi-modal data that capture their spatio-temporal correlation, (ii) provide sufficient conditions for the identifiability of nth order cumulants and also for a special class of heavy tailed distributions. Further, we investigate conditions on network routing for the flows that prove sufficient for identifiability of their distributions (up to mean). Finally, we extend our results to directed acyclic graphs and discuss some open problems.

Effectiveness of Bootstrap Bias Correction in the Context of Clock Offset Estimators

Estimating and correcting the offset between two or more clocks is an important problem in data communication networks. For example, Internet telephony depends on network routers having a common notion of time, and cellular networks provide a higher quality of service by using transmission protocols that depend on neighboring base stations knowing the offset that exists between their local clocks. In previous work it was shown that bootstrap bias correction of Paxson's well-known estimator of clock offset produces an estimator with improved bias and mean squared error (MSE) properties. In addition, the ordered-BLUE (o-BLUE) under an exponential distribution for network delays was derived, and its bootstrap bias-corrected form was shown to have lower bias and MSE than the bootstrap bias-corrected form of Paxson's estimator when network delays follow lognormal, gamma, and Weibull distributions. The inferred robustness of the bias-corrected o-BLUE to the assumed distribution of network delays is an attractive property, because no single family of distributions can consistently characterize network delays. Recent Internet traffic modeling research has suggested that the Pareto distribution is an applicable distribution for network delays. That finding motivates the work in this article concerned with clock offset estimation and the effectiveness of bootstrap bias-corrected estimators in the context of heavy-tailed network delays. An important result is the demonstrated robustness of the bias-corrected form of Paxson's estimator in the presence of heavy-tailed network delays. An additional and somewhat surprising result is that if network delays follow a Pareto distribution, then bootstrap bias correction of the exponential o-BLUE fails in the sense that the absolute bias increases. The same finding is observed in an alternative class of heavy-tailed distributions, where the success or failure of bootstrap bias correction is traceable to a parameter that reflects the weight of the tail. Although it is well known that bias correction can increase MSE, to the best of our knowledge no practical application in which it increases the absolute bias has been identified up to this point.