Heavy-tailed Models Research Articles

Is Seeing Believing? A Practitioner's Perspective on High-Dimensional Statistical Inference in Cancer Genomics Studies.

Variable selection methods have been extensively developed for and applied to cancer genomics data to identify important omics features associated with complex disease traits, including cancer outcomes. However, the reliability and reproducibility of the findings are in question if valid inferential procedures are not available to quantify the uncertainty of the findings. In this article, we provide a gentle but systematic review of high-dimensional frequentist and Bayesian inferential tools under sparse models which can yield uncertainty quantification measures, including confidence (or Bayesian credible) intervals, p values and false discovery rates (FDR). Connections in high-dimensional inferences between the two realms have been fully exploited under the "unpenalized loss function + penalty term" formulation for regularization methods and the "likelihood function × shrinkage prior" framework for regularized Bayesian analysis. In particular, we advocate for robust Bayesian variable selection in cancer genomics studies due to its ability to accommodate disease heterogeneity in the form of heavy-tailed errors and structured sparsity while providing valid statistical inference. The numerical results show that robust Bayesian analysis incorporating exact sparsity has yielded not only superior estimation and identification results but also valid Bayesian credible intervals under nominal coverage probabilities compared with alternative methods, especially in the presence of heavy-tailed model errors and outliers.

On Models of Time Series

Recently a number of time series models have been constructed and studied for real valued observations using different marginal distributions by several researchers. Many naturally occurring time series observations show a tendency to follow asymmetric and heavy tailed distributions. A review of literature on some heavy tailed models of time series is done in this paper.

A heavy-tailed model for analyzing miRNA-seq raw read counts.

This article addresses the limitations of existing statistical models in analyzing and interpreting highly skewed miRNA-seq raw read count data that can range from zero to millions. A heavy-tailed model using discrete stable distributions is proposed as a novel approach to better capture the heterogeneity and extreme values commonly observed in miRNA-seq data. Additionally, the parameters of the discrete stable distribution are proposed as an alternative target for differential expression analysis. An R package for computing and estimating the discrete stable distribution is provided. The proposed model is applied to miRNA-seq raw counts from the Norwegian Women and Cancer Study (NOWAC) and the Cancer Genome Atlas (TCGA) databases. The goodness-of-fit is compared with the popular Poisson and negative binomial distributions, and the discrete stable distributions are found to give a better fit for both datasets. In conclusion, the use of discrete stable distributions is shown to potentially lead to more accurate modeling of the underlying biological processes.

The Bayesian regularized quantile varying coefficient model

The quantile varying coefficient (VC) model can flexibly capture dynamical patterns of regression coefficients. In addition, due to the quantile check loss function, it is robust against outliers and heavy-tailed distributions of the response variable, and can provide a more comprehensive picture of modeling via exploring the conditional quantiles of the response variable. Although extensive studies have been conducted to examine variable selection for the high-dimensional quantile varying coefficient models, the Bayesian analysis has been rarely developed. The Bayesian regularized quantile varying coefficient model has been proposed to incorporate robustness against data heterogeneity while accommodating the non-linear interactions between the effect modifier and predictors. Selecting important varying coefficients can be achieved through Bayesian variable selection. Incorporating the multivariate spike-and-slab priors further improves performance by inducing exact sparsity. The Gibbs sampler has been derived to conduct efficient posterior inference of the sparse Bayesian quantile VC model through Markov chain Monte Carlo (MCMC). The merit of the proposed model in selection and estimation accuracy over the alternatives has been systematically investigated in simulation under specific quantile levels and multiple heavy-tailed model errors. In the case study, the proposed model leads to identification of biologically sensible markers in a non-linear gene-environment interaction study using the NHS data.

Forecasting of symmetric [formula omitted]stable autoregressive models by time series approach supported by artificial neural networks

Recent research activities in forecasting suggest that artificial neural networks can be a promising alternative to the traditional linear models. However, no single model, either linear or nonlinear is capable of obtaining the forecasts accurately. In this paper, a hybrid methodology that combines symmetric α-stable autoregressive time series and artificial neural networks is proposed. The methodology is validated through Monte-Carlo simulations. Moreover, the new method is used to model real empirical data thus showing the usefulness of heavy-tailed models supported by artificial neural networks in statistical modeling.

Heavy Tails and the Shape of Modified Numerals.

The pattern of implicatures of the modified numeral “more than n” depends on the roundness of n. Cummins et al. (2012) present experimental evidence for the relation between roundness and implicature patterns and propose a pragmatic account of the phenomenon. More recently, Hesse and Benz (2020) present more extensive evidence showing that implicatures also depend on the magnitude of n and propose a novel explanation based on the approximate number system (Dehaene, 1999). Despite the wealth of experimental data, no formal account has yet been proposed to characterize the full posterior distribution over numbers of a listener after hearing “more than n.” We develop one such account within the Rational Speech Act framework, quantitatively reconstructing the pragmatic reasoning of a rational listener. We argue that world knowledge about the distribution of the true quantity has a substantial impact on the information conveyed by the modified numeral. We show that our pragmatic account in combination with a heavy‐tailed model of the participants' prior correctly predicts various features of the experimental data from Hesse and Benz (2020).

Portfolio Optimization for Extreme Risks with Maximum Diversification: An Empirical Analysis

Heavy tailedness and interconnectedness widely exist in stock returns and large insurance claims, which contributes to huge losses for financial institutions. Diversification ratio (DR) measures the degree of diversification using the Value-at-Risk, which is known to capture extreme risks better than variance. The portfolio optimization strategy based on DR maximizes the effect of diversification for extreme risks. In this paper, we empirically examine the DR strategy by using more than 350 S&P 500 stocks under the assumption that the stock losses are modeled with a flexible multivariate heavy-tailed model. This assumption is verified empirically. The performance of DR strategy is compared with four benchmark strategies: equally weighted portfolio, minimum-variance portfolio, extreme risk index portfolio, and most diversified portfolio. The performance of comparison includes annualized portfolio return, modified Sharpe ratio, maximum drawdown, portfolio concentration, portfolio turnover, and the degree of diversification. DR outperforms other strategies. In particular, DR shows the highest return and maintains the highest level of diversification during the global financial crisis of 2007–2009.

Fast inference for robust nonlinear mixed-effects models

The interest for nonlinear mixed-effects models comes from application areas as pharmacokinetics, growth curves and HIV viral dynamics. However, the modeling procedure usually leads to many difficulties, as the inclusion of random effects, the estimation process and the model sensitivity to atypical or nonnormal data. The scale mixture of normal distributions include heavy-tailed models, as the Student-t, slash and contaminated normal distributions, and provide competitive alternatives to the usual models, enabling the obtention of robust estimates against outlying observations. Our proposal is to compare two estimation methods in nonlinear mixed-effects models where the random components follow a multivariate scale mixture of normal distributions. For this purpose, a Monte Carlo expectation-maximization algorithm (MCEM) and an efficient likelihood-based approximate method are developed. Results show that the approximate method is much faster and enables a fairly efficient likelihood maximization, although a slightly larger bias may be produced, especially for the fixed-effects parameters. A discussion on the robustness aspects of the proposed models are also provided. Two real nonlinear applications are discussed and a brief simulation study is presented.

Heavy tailed modeling of automobile claim data from Ghana

Africa has long been ignored with respect to among others insurance modeling. In this paper, we propose a model for automobile claim data from Ghana. The body of the data are modeled by a lognormal distribution. However, the tail is noted be too heavy to be modeled by a single heavy tailed distribution. A mixture of distributions is used to model the tail. Estimates of risk are given.

Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees

We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d+1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures, describing localization at different heights. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. We show that the two conditions can be fulfilled at the same time, which then implies coexistence of both types of measures for large classes of models including the SOS-model, and heavy-tailed models arising for instance for potentials of logarithmic growth.

Causal discovery in heavy-tailed models

Causal questions are omnipresent in many scientific problems. While much progress has been made in the analysis of causal relationships between random variables, these methods are not well suited if the causal mechanisms only manifest themselves in extremes. This work aims to connect the two fields of causal inference and extreme value theory. We define the causal tail coefficient that captures asymmetries in the extremal dependence of two random variables. In the population case, the causal tail coefficient is shown to reveal the causal structure if the distribution follows a linear structural causal model. This holds even in the presence of latent common causes that have the same tail index as the observed variables. Based on a consistent estimator of the causal tail coefficient, we propose a computationally highly efficient algorithm that estimates the causal structure. We prove that our method consistently recovers the causal order and we compare it to other well-established and nonextremal approaches in causal discovery on synthetic and real data. The code is available as an open-access R package.

Yule–Walker Equations Using a Gini Covariance Matrix for the High-Dimensional Heavy-Tailed PVAR Model

Gini covariance plays a vital role in analyzing the relationship between random variables with heavy-tailed distributions. In this papaer, with the existence of a finite second moment, we establish the Gini–Yule–Walker equation to estimate the transition matrix of high-dimensional periodic vector autoregressive (PVAR) processes, the asymptotic results of estimators have been established. We apply this method to study the Granger causality of the heavy-tailed PVAR process, and the results show that the robust transfer matrix estimation induces sign consistency in the value of Granger causality. Effectiveness of the proposed method is verified by both synthetic and real data.

Foundations & Quantitative Aspects of Operational Risk Modelling

Presentation on fundamentals of Operational Risk Modelling. The core components of a quantitative operational risk modelling framework. Based on references: 1. Cruz MG, Peters GW, Shevchenko PV. Fundamental aspects of operational risk and insurance analytics: A handbook of operational risk. John Wiley & Sons; 2015 Jan 29. 2. Peters GW, Shevchenko PV. Advances in heavy tailed risk modeling: A handbook of operational risk. John Wiley & Sons; 2015 May 5.

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.

Random ecological networks that depend on ephemeral wetland complexes

Wetlands provide diverse ecosystem services: they create habitats, provide hydrological control, and regulate biogeochemical cycles of flora and fauna through interaction with surrounding wetlands. These services vary temporally, depending primarily on hydro-climatic conditions and the type of wetland. To better understand the functions of wetlands, it is necessary to analyze the effect of stochastic hydro-climatic conditions on the dynamics of wetland surface area and the resulting dispersal of species. In this study, we analyzed the dynamics of the connectivity and efficiency of ecological networks according to (1) the characteristics of the wetland area distribution, which varies by season, and (2) dispersal models (threshold distance, exponential kernel, and heavy-tailed movement). The results indicate that hydrologically sensitive small wetlands were often the primary elements that characterize wetland area distribution. Consequently, during dry seasons, the corresponding ecological network topology, measured by mean degree, network efficiency, and clustering coefficient decreased in all dispersal models we tested. We also found that while the correlation of hubs between dispersal models was strong, the levels of network vulnerability, evaluated by removing those hubs, were different, with the heavy-tailed model showing the highest vulnerability. Moreover, we observed the possibility of regime shift in ecological networks when a chronic, high dryness induced significant reduction of large wetlands. Our network modeling approach based on hydrologic systems will provide a new decision-making process and perspectives for conservation and restoration planning including the conditions regarding climate change.

Robust estimation of single index models with responses missing at random

A single-index regression model is considered, where some responses in the model are assumed to be missing at random. Local linear rank-based estimators of the single-index direction and the unknown link function are proposed. Asymptotic properties of the estimators are established under mild regularity conditions. Monte Carlo simulation experiments show that the proposed estimators are more efficient than their least squares counterparts especially when the data are derived from contaminated or heavy-tailed model error distributions. When the errors follow a normal distribution, the least squares index direction estimator tends to be more efficient than the rank-based index direction estimator; however, the least squares link function estimator remains less efficient than the rank-based link function estimator. A real data example is analyzed and cross-validation studies show that the proposed procedure provides better prediction than the least squares method when the responses contain outliers and are missing at random.

Alternative heavy tailed models in seismology

Great earthquakes are commonly considered as the ones with moment magnitude (Mw ) above or equal to 8.0. Since these earthquakes can destroy entire communities located near the epicentre, the search of physical laws that explain the energy released by them is an important issue. There is a connection between the radiated energy of an earthquake, its magnitude and its seismic moment (M 0). Thence, when fitting a heavy or an extremely heavy tailed distribution to a seismic moment dataset, we are in fact adjusting a mathematical model which explains the amount of energy released by these great seisms. Therefore, the main goal of this work is to study the more appropriated Pareto based models (the most used family in this field) when explaining the seismic moment of the great earthquakes. With this purpose in mind, we selected two different catalogs that accommodate recent events and are considered more accurate than other catalogs used in previous works. We conclude that the traditional Pareto distribution remains a good choice to deal with this kind of data, but Log-Pareto lead to higher p-values and Location-scale Pareto is better fitted to the biggest events.

On the comparison of several classical estimators of the extreme value index

Due to the fact that for heavy tails the classical Hill estimator of a positive extreme value index is asymptotically biased, new and interesting alternative estimators have appeared in the literature. In this work we compare several classical estimators of the extreme value index based on moments of the upper order statistics. Since several alternative estimators have eventually a null asymptotic bias, for some heavy tailed models, the comparison is performed not only with the Hill and recent generalized means estimators but also with an asymptotically unbiased Hill estimator. The comparison study is performed asymptotically, under a third-order framework, and for finite samples, through a Monte Carlo simulation study.

A new family of heavy tailed distributions with an application to the heavy tailed insurance loss data

Heavy tailed distributions play very significant role in the study of actuarial and financial risk management data but the probability distributions proposed to model such data are scanty. Actuaries often search for new and appropriate statistical models to address data related to financial risk problems. In this work, we propose a new family of heavy tailed distributions. Some basic properties of this new family of heavy tailed distributions are obtained. A special sub-model of the proposed family, called a new heavy tailed Weibull model is considered in detail. The maximum likelihood estimators of the model parameters are obtained. A Monte Carlo simulation study is carried out to evaluate the performance of these estimators. Furthermore, some actuarial measures such as value at risk and tail value at risk are calculated. A simulation study based on these actuarial measures is done. Finally, an application of the proposed model to a heavy tailed insurance loss data set is presented.

Separation of alpha-stable random vectors

Source separation aims at decomposing a vector into additive components. This is often done by first estimating source parameters before feeding them into a filtering method, often based on ratios of covariances. The whole pipeline is traditionally rooted in some probabilistic framework providing both the likelihood for parameter estimation and the separation method. While Gaussians are ubiquitous for this purpose, many studies showed the benefit of heavy-tailed models for estimation. However, there is no counterpart filtering method to date exploiting such formalism, so that related studies revert to covariance-based filtering after estimation is finished. Here, we introduce a new multivariate separation technique, that fully exploits the flexibility of α-stable heavy-tailed distributions. We show how a spatial representation can be exploited, which decomposes the observation as an infinite sum of contributions originating from all directions. Two methods for separation are derived. The first one is non-linear and similar to a beamforming technique, while the second one is linear, but minimizes a covariation criterion, which is the counterpart of the covariance for α-stable vectors. We evaluate the proposed techniques in a large number of challenging and adverse situations on synthetic experiments, demonstrating their performance for the extraction of signals from strong interferences.