Heavy-tailed Research Articles

Randomly Stopped Sums with Generalized Subexponential Distribution

Let {ξ1,ξ2,…} be a sequence of independent possibly differently distributed random variables, defined on a probability space (Ω,F,P) with distribution functions {Fξ1,Fξ2,…}. Let η be a counting random variable independent of sequence {ξ1,ξ2,…}. In this paper, we find conditions under which the distribution function of randomly stopped sum Sη=ξ1+ξ2+…+ξη belongs to the class of generalized subexponential distributions.

Non-Markovian epidemic spreading on temporal networks

Many empirical studies have revealed that the occurrences of contacts associated with human activities are non-Markovian temporal processes with a heavy tailed inter-event time distribution. Besides, there has been increasing empirical evidence that the infection and recovery rates are time-dependent. However, we lack a comprehensive framework to analyze and understand non-Markovian contact and spreading processes on temporal networks. In this paper, we propose a general formalism to study non-Markovian dynamics on non-Markovian temporal networks. We find that, under certain conditions, non-Markovian dynamics on temporal networks are equivalent to Markovian dynamics on static networks. Interestingly, this result is independent of the underlying network topology.

Spearman Rank Correlation Screening for Ultrahigh-Dimensional Censored Data

Herein, we propose a Spearman rank correlation-based screening procedure for ultrahigh-dimensional data with censored response cases. The proposed method is model-free without specifying any regression forms of predictors or response variables and is robust under the unknown monotone transformations of these response variable and predictors. The sure-screening and rank-consistency properties are established under some mild regularity conditions. Simulation studies demonstrate that the new screening method performs well in the presence of a heavy-tailed distribution, strongly dependent predictors or outliers, and offers superior performance over the existing nonparametric screening procedures. In particular, the new screening method still works well when a response variable is observed under a high censoring rate. An illustrative example is provided.

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.

Probability Density Estimation through Nonparametric Adaptive Partitioning and Stitching

We present a novel nonparametric adaptive partitioning and stitching (NAPS) algorithm to estimate a probability density function (PDF) of a single variable. Sampled data is partitioned into blocks using a branching tree algorithm that minimizes deviations from a uniform density within blocks of various sample sizes arranged in a staggered format. The block sizes are constructed to balance the load in parallel computing as the PDF for each block is independently estimated using the nonparametric maximum entropy method (NMEM) previously developed for automated high throughput analysis. Once all block PDFs are calculated, they are stitched together to provide a smooth estimate throughout the sample range. Each stitch is an averaging process over weight factors based on the estimated cumulative distribution function (CDF) and a complementary CDF that characterize how data from flanking blocks overlap. Benchmarks on synthetic data show that our PDF estimates are fast and accurate for sample sizes ranging from 29 to 227, across a diverse set of distributions that account for single and multi-modal distributions with heavy tails or singularities. We also generate estimates by replacing NMEM with kernel density estimation (KDE) within blocks. Our results indicate that NAPS(NMEM) is the best-performing method overall, while NAPS(KDE) improves estimates near boundaries compared to standard KDE.

An efficient implementation of a conservative finite volume scheme with constant and linear reconstructions for solving the coagulation equation

Population balances provide an economic means of describing dense particulate phases with particle-particle interactions. Here, we focus on particle coagulation and address the evaluation of the corresponding integral source terms within the scope of a conservative finite volume formulation. Based on one kernel evaluation for every pair of parent cells, efficient analytical formulas are derived that obviate the explicit decomposition of the integration domains into elementary geometrical shapes and are valid for arbitrary volume-grids. Considering the volume-weighted cell averages as degrees of freedom, the formulas are presented for both cell-wise constant and linear reconstructions of the particle volume distribution. We find the latter to be effective at mitigating the unphysically heavy tails that are characteristic of solutions obtained with piecewise constant reconstructions. For Brownian coagulation, the computational expense is mainly caused by the kernel evaluations, while the cost of the parent-daughter pairing and integration algorithm is almost negligibly small.

Robust penalized empirical likelihood in high dimensional longitudinal data analysis

As an effective nonparametric method, empirical likelihood (EL) is appealing in combining estimating equations flexibly and adaptively for incorporating data information. To select important variables and estimating equations in a sparse high-dimensional model, a penalized EL method based on robust estimating functions is proposed for regularizing the regression parameters and the associated Lagrange multipliers simultaneously, which allows the dimensionality of both regression parameters and estimating equations to grow exponentially with the sample size. A first inspection on the robustness of estimating equations contributing to variable selection is discussed from both theoretical perspective and intuitive simulation results in this paper. The proposed method can improve the robustness and effectiveness when the data have underlying outliers or heavy tails in the response variables and/or covariates. The robustness of the estimator is measured via the bounded influence function, and the oracle properties are also established under some regularity conditions. Extensive simulation studies and yeast cell data are used to evaluate the performance of the proposed method. The numerical results reveal that the robustness of sparse estimating equations selection fundamentally enhances variable selection accuracy when the data have heavy tails and/or include underlying outliers.

Finite-Time Ruin Probabilities of Bidimensional Risk Models with Correlated Brownian Motions

The present work concerns the finite-time ruin probabilities for several bidimensional risk models with constant interest force and correlated Brownian motions. Under the condition that the two Brownian motions {B1(t),t≥0} and {B2(t),t≥0} are correlated, we establish new results for the finite-time ruin probabilities. Our research enriches the development of the ruin theory with heavy tails in unidimensional risk models and the dependence theory of stochastic processes.

Robust estimation for dynamic single index varying coefficient models

In this article, we propose a new estimation procedure based on modal regression for dynamic single index varying coefficient models, where the index coefficient functions and the link functions are approximated by B-spline basis, respectively. Under some mild regularity conditions, we establish the asymptotic normalities of the obtained spline-based estimators. By introducing an additional tuning parameter (e.g. h), the proposed method can produce robust estimates when the data are not well behaved (e.g. in the presence of outliers and/or heavy-tailed error distributions), and we further calculate the robust estimates by using a modified expectation-maximization-type iteration algorithm, and develop a data driven procedure to select the tuning parameters for the proposed approaches. Finally, some simulations and a real data application are conducted to illustrate the utility of the proposed methodology.

Retire: Robust expectile regression in high dimensions

High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted ℓ1-penalization which reduces the estimation bias from ℓ1-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which d≪n; (ii) high-dimensional regime in which s≪n≪d with s denoting the number of significant predictors. In the high-dimensional setting, we thoroughly analyze the statistical properties of the solution path of iteratively reweighted ℓ1-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we demonstrate that with as few as log(logd) iterations, the final iterate of our proposed approach achieves the oracle convergence rate. At each iteration, we solve the weighted ℓ1-penalized convex program using a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the promising performance of the proposed procedure in comparison to both non-robust and quantile regression based alternatives.

Inverse Hamza Distribution: Properties and Applications to Lifetime Data

In this paper, a new distribution, named ‘the Inverted Hamza distribution’, was introduced. It is an extension of the Hamza distribution that can model real-world data with an upside-down bathtub shape and heavy tails. Mathematical and statistical characteristics such as the quantile function, moments, entropy measure, stochastic ordering and distribution of order statistics have been derived. Furthermore, dependability measures such as the survival function and hazard function have been developed. The greatest likelihood technique was used to estimate the distribution parameters. To demonstrate the applicability of the distribution, a numerical example was given. According on the results, the proposed distribution outperformed the competing distributions.

Weighted l1‐Penalized Corrected Quantile Regression for High‐Dimensional Temporally Dependent Measurement Errors

This article derives some large sample properties of weighted ‐penalized corrected quantile estimators of the regression parameter vector in a high‐dimensional errors in variables (EIVs) linear regression model. In this model, the number of predictors depends on the sample size and tends to infinity, generally at a faster rate than , as tends to infinity. Moreover, the measurement errors in the covariates are assumed to have linear stationary temporal dependence and known Laplace marginal distribution while the regression errors are assumed to be independent non‐identically distributed random variables having possibly heavy tails. The article discusses some rates of consistency of these estimators, a model consistency result and an appropriate data adaptive algorithm for obtaining a suitable choice of weights. A simulation study assesses the finite sample performance of some of the proposed estimators. This article also contains analogs of Massart's inequality for independent and short memory moving average predictors, which is instrumental in establishing the said consistency rates of the above mentioned estimators in the current setup of high dimensional EIVs regression models.

Score correlation

Based on the new concept of the scalar-valued score function of continuous distributions we introduce the score correlation coefficient “tailored” to the assumed probabilistic model. By means of simulation experiments we show that it can reveal association of skewed heavy-tailed distributions with high kurtosis.

Catoni-style confidence sequences for heavy-tailed mean estimation

A confidence sequence (CS) is a sequence of confidence intervals that is valid at arbitrary data-dependent stopping times. These are useful in applications like A/B testing, multi-armed bandits, off-policy evaluation, election auditing, etc. We present three approaches to constructing a confidence sequence for the population mean, under the minimal assumption that only an upper bound σ2 on the variance is known. While previous works rely on light-tail assumptions like boundedness or subGaussianity (under which all moments of a distribution exist), the confidence sequences in our work are able to handle data from a wide range of heavy-tailed distributions. The best among our three methods — the Catoni-style confidence sequence — performs remarkably well in practice, essentially matching the state-of-the-art methods for σ2-subGaussian data, and provably attains the loglogt/t lower bound due to the law of the iterated logarithm. Our findings have important implications for sequential experimentation with unbounded observations, since the σ2-bounded-variance assumption is more realistic and easier to verify than σ2-subGaussianity (which implies the former). We also extend our methods to data with infinite variance, but having pth central moment (1<p<2).

Study on Bayesian Skew-Normal Linear Mixed Model and Its Application in Fire Insurance.

Fire insurance is a crucial component of property insurance, and its rating depends on the forecast of insurance loss claim data. Fire insurance loss claim data have complicated characteristics such as skewness and heavy tail. The traditional linear mixed model is commonly difficult to accurately describe the distribution of loss. Therefore, it is crucial to establish a scientific and reasonable distribution model of fire insurance loss claim data. In this study, the random effects and random errors in the linear mixed model are firstly assumed to obey the skew-normal distribution. Then, a skew-normal linear mixed model is established using the Bayesian MCMC method based on a set of U.S. property insurance loss claims data. Comparative analysis is conducted with the linear mixed model of logarithmic transformation. Afterward, a Bayesian skew-normal linear mixed model for Chinese fire insurance loss claims data is designed. The posterior distribution of claim data parameters and related parameter estimation are employed with the R language JAGS package to obtain the predicted and simulated loss claim values. Finally, the optimization model in this study is used to determine the insurance rate. The results demonstrate that the model established by the Bayesian MCMC method can overcome data skewness, and the fitting and correlation with the sample data are better than the log-normal linear mixed model. Hence, it can be concluded that the distribution model proposed in this paper is reasonable for describing insurance claims. This study innovates a new approach for calculating the insurance premium rate and expands the application of the Bayesian method in the fire insurance field.

Nonparametric inference about increasing odds rate distributions

To improve nonparametric estimates of lifetime distributions, we propose using the increasing odds rate (IOR) model as an alternative to other popular, but more restrictive, ‘adverse ageing’ models, such as the increasing hazard rate one. This extends the scope of applicability of some methods for statistical inference under order restrictions, since the IOR model is compatible with heavy-tailed and bathtub distributions. We study a strongly uniformly consistent estimator of the cumulative distribution function of interest under the IOR constraint. Numerical evidence shows that this estimator often outperforms the classic empirical distribution function when the underlying model does belong to the IOR family. We also study two different tests to detect deviations from the IOR property and establish their consistency. The performance of these tests is also evaluated through simulations.

Dynamic train dwell time forecasting: a hybrid approach to address the influence of passenger flow fluctuations

AbstractTrain timetables and operations are defined by the train running time in sections, dwell time at stations, and headways between trains. Accurate estimation of these factors is essential to decision-making for train delay reduction, train dispatching, and station capacity estimation. In the present study, we aim to propose a train dwell time model based on an averaging mechanism and dynamic updating to address the challenges in the train dwell time prediction problem (e.g., dynamics over time, heavy-tailed distribution of data, and spatiotemporal relationships of factors) for real-time train dispatching. The averaging mechanism in the present study is based on multiple state-of-the-art base predictors, enabling the proposed model to integrate the advantages of the base predictors in addressing the challenges in terms of data attributes and data distributions. Then, considering the influence of passenger flow on train dwell time, we use a dynamic updating method based on exponential smoothing to improve the performance of the proposed method by considering the real-time passenger amount fluctuations (e.g., passenger soars in peak hours or passenger plunges during regular periods). We conduct experiments with the train operation data and passenger flow data from the Chinese high-speed railway line. The results show that due to the advantages over the base predictors, the averaging mechanism can more accurately predict the dwell time at stations than its counterparts for different prediction horizons regarding predictive errors and variances. Further, the experimental results show that dynamic smoothing can significantly improve the accuracy of the proposed model during passenger amount changes, i.e., 15.4% and 15.5% corresponding to the mean absolute error and root mean square error, respectively. Based on the proposed predictor, a feature importance analysis shows that the planned dwell time and arrival delay are the two most important factors to dwell time. However, planned time has positive influences, whereas arrival delay has negative influences.

GNSS integrity risk evaluation in the position domain based on the generalized Pareto distribution

Integrity monitoring of global navigation satellite systems (GNSSs) is designed to protect against extremely rare hazardous events, characterized by an integrity risk with a very low probability. The traditional integrity risk evaluation is restricted simultaneously by non-Gaussian measurement errors and impractical time consumption. Based on extreme value theory, a generalized Pareto distribution (GPD)-based integrity risk evaluation method in the position domain is proposed to estimate the upper bound of the integrity risk. In order to account for the GPD modeling error and estimation error, conservatism of the proposed GPD-based integrity risk evaluation is obtained by imposing model-driven and data-driven overbounding. Simulation results from four typical heavy-tailed distributions have shown that conservative and tight bound integrity risk results can be achieved. Furthermore, real-world European Geostationary Navigation Overlay Service measurements experiment has shown that the integrity risk evaluation resulting from the proposed method is at least one order less than the traditional evaluation method, which is consistent with official publications.

A new multivariate control chart based on the isolation forest algorithm

Statistical process control (SPC) is an extensively utilized technique in quality management for service and manufacturing industries. Traditional SPC control charts rely on normality assumptions and analyze data in one dimension. Machine learning methods have been increasingly used to address limitations of traditional SPC techniques, such as one-class classification (OCC) procedures, using support vector machine (SVM), K-nearest neighbor (KNN), and other techniques. Nonetheless, there are drawbacks associated with these methods. For instance, SVM-based approaches may require significant computational resources, whereas KNN-based approaches may encounter difficulties in identifying minor variations. Here, these limitations are addressed using a new SPC framework that incorporates the isolation forest (iForest) algorithm, an unsupervised anomaly detection technique developed for machine learning applications. Our presented method examines how the number of trees impacts the charting scheme and offers several advantages: (1) it has a low constant time complexity that results in fast computing speeds; (2) it outperforms competitors in detecting small shifts in various distributions; and (3) while dealing with challenging distributions with heavy tails, it proves to be effective. We provide two examples that demonstrate the performance of our proposed control chart using Monte Carlo simulations.

A general unit hydrograph distribution and its application on the marginal distribution of global wind speed

We here introduce a simple, general, and analytical method based on the general unit hydrograph (GUH) theory proposed by Guo (2022a, 2022b, 2022c) to describe wind speed marginal distribution. Wind speed probability is generally described by the Weibull distribution; however, its accuracy depends on the contribution made by null wind speed (owing to its zero initial condition). To overcome this disadvantage, we propose a three-parameter GUH distribution with significant flexibility in modeling symmetric heavy-tailed distributions as well as skewed distributions. We show that our proposed GUH distribution corresponds to a generalized logistic distribution that contains both type I and type II generalized logistic distributions. The shape properties of the GUH distribution are discussed along with a simple method for estimating parameters by means of least squares methods. The flexibility of the proposed GUH distribution was assessed by applying it to a well reconstructed global gridded wind speed dataset. Subsequently, we quantify the changes caused by a reversal in global surface wind speed stilling using our GUH distribution. Finally, we confirm that our proposed GUH model agrees with reconstructed wind speed data better than conventional Weibull distribution models, which suggests that the new distribution can be a useful tool for sustainable development of wind energy.