Heavy-tailed Research Articles

A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data

The half-normal distribution is composited with the Pareto model to obtain a uni-parametric distribution with a heavy right tail, called the composite half-normal-Pareto distribution. This new distribution is useful for modeling positive data with atypical observations. We study the properties and the behavior of the right tail of this new distribution. We estimate the parameter using a method based on percentiles and the maximum likelihood method and assess the performance of the maximum likelihood estimator using Monte Carlo. We report three applications, one with simulated data and the others with income and expenditure data, in which the new distribution presents better performance than the Pareto distribution.

Precise large deviations for sums of dependent random variables with subexponential distribution

. In this article, we study the precise large deviations for sums of non identically distributed random variables { X i , i ≥ 1 } with distribution functions { F i , i ≥ 1 } that are widely upper orthant dependent (WUOD, in short) and ϕ-mixing. Moreover, the random variables are dependent, according to some copulas. We assume that the average of the right tails of distribution functions F i is equivalent to the subexponential distribution F.

The Lomax-Exponentiated Odds Ratio–G Distribution and Its Applications

This paper introduces the Lomax-exponentiated odds ratio–G (L-EOR–G) distribution, a novel framework designed to adeptly navigate the complexities of modern datasets. It blends theoretical rigor with practical application to surpass the limitations of traditional models in capturing complex data attributes such as heavy tails, shaped curves, and multimodality. Through a comprehensive examination of its theoretical foundations and empirical data analysis, this study lays down a systematic theoretical framework by detailing its statistical properties and validates the distribution’s efficacy and robustness in parameter estimation via Monte Carlo simulations. Empirical evidence from real-world datasets further demonstrates the distribution’s superior modeling capabilities, supported by compelling various goodness-of-fit tests. The convergence of theoretical precision and practical utility heralds the L-EOR–G distribution as a groundbreaking advancement in statistical modeling, significantly enhancing precision and adaptability. The new model not only addresses a critical need within statistical modeling but also opens avenues for future research, including the development of more sophisticated estimation methods and the adaptation of the model for various data types, thereby promising to refine statistical analysis and interpretation across a wide array of disciplines.

Uniform asymptotics for a risk model with constant force of interest and a random number of delayed claims

Consider an insurance risk model with constant force of interest under the assumption that an immediate claim is accompanied by a random number of delayed claims. In the presence of heavy tails on claim distributions and dependence structures among modelling components, we study the uniform asymptotics for an insurer's ruin-related quantities, where the uniformity holds for all time horizons varying in a relevant interval.

Online updating mode learning for streaming datasets

In the case of limited computer memory, it is worth discussing how to regression the big data stream and solve the outlier problem reasonably. To solve the above problems, this article proposes an online updatable modal linear regression algorithm, which is robust to heavy-tailed distribution, heterogeneous errors and outliers. However, due to streaming data, the traditional static statistical methods are facing new challenges. In addition, existing inference tools in online learning cannot be used directly for modal linear regression. A major challenge is that the data is one-pass and the final amount of data is not even known, which results in the inability to obtain optimal bandwidth, resulting in poor theoretical performance of the estimator. In particular, the estimation and inference methods are proposed. The process only passes the data once. Finally, the finite sample performance of the proposed method is verified by simulation and real data analysis.

Loss modeling with the size-biased lognormal mixture and the entropy regularized EM algorithm

The Erlang mixture with a common scale parameter is one of many popular models for modeling insurance losses. However, the actuarial literature recognizes and discusses some limitations of aforementioned model in approximate heavy-tailed distributions. In this paper, a size-biased left-truncated Lognormal (SB-ltLN) mixture is proposed as a robust alternative to the Erlang mixture for modeling left-truncated insurance losses with a heavy tail. The weak denseness property of the weighted Lognormal mixture is studied along with the tail behavior. Explicit analytical solutions are derived for moments and Tail Value at Risk based on the proposed model. An extension of the regularized expectation–maximization (REM) algorithm with Shannon's entropy weights (ewREM) is introduced for parameter estimation and variability assessment. The Operational Riskdata eXchange's left-truncated internal fraud loss data set is used to illustrate applications of the proposed model. Finally, the results of a simulation study show promising performance of the proposed SB-ltLN mixture in different simulation settings.

Heavy tail distribution and Deuterium excess for drought assessment case of Djelfa- watershed (Algeria)

Heavy tail distribution and Deuterium excess for drought assessment case of Djelfa- watershed (Algeria)

Reduced-bias estimation of the extreme conditional tail expectation for Box–Cox transforms of heavy-tailed distributions

Conditional tail expectation (CTE) is a coherent risk measure defined as the mean of the loss distribution above a high quantile. The existence of the CTE as well as the asymptotic properties of associated estimators however require integrability conditions that may be violated when dealing with heavy-tailed distributions. We introduce Box–Cox transforms of the CTE that have two benefits. First, they alleviate these theoretical issues. Second, they enable to recover a number of risk measures such as conditional tail expectation, expected shortfall, conditional value-at-risk or conditional tail variance. The construction of dedicated estimators is based on the investigation of the asymptotic relationship between Box–Cox transforms of the CTE and quantiles at extreme probability levels, as well as on an extrapolation formula established in the heavy-tailed context. We quantify and estimate the bias induced by the use of these approximations and then introduce reduced-bias estimators whose asymptotic properties are rigorously shown. Their finite-sample properties are assessed on a simulation study and illustrated on real data, highlighting the practical interest of both the bias reduction and the Box–Cox transform.

Anomalous diffusion of synaptic vesicles and its influences on spontaneous and evoked neurotransmission.

Neurons in the central nervous system communicate with each other by activating billions of tiny synaptic boutons distributed along their fine axons. These presynaptic varicosities are very crowded environments, comprising hundreds of synaptic vesicles. Only a fraction of these vesicles can be recruited in a single release episode, either spontaneous or evoked by action potentials. Since the seminal work by Fatt and Katz, spontaneous release has been modelled as a memoryless process. Nevertheless, at central synapses, experimental evidence indicates more complex features, including non-exponential distributions of release intervals and power-law behaviour in their rate. To describe these features, we developed a probabilistic model of spontaneous release based on Brownian motion of synaptic vesicles in the presynaptic environment. To account for different diffusion regimes, we based our simulations on fractional Brownian motion. We show that this model can predict both deviation from the Poisson hypothesis and power-law features in experimental quantal release series, thus suggesting that the vesicular motion by diffusion could per se explain the emergence of these properties. We demonstrate the efficacy of our modelling approach using electrophysiological recordings at single synaptic boutons and ultrastructural data. When this approach was used to simulate evoked responses, we found that the replenishment of the readily releasable pool driven by Brownian motion of vesicles can reproduce the characteristic binomial release distributions seen experimentally. We believe that our modelling approach supports the idea that vesicle diffusion and readily releasable pool dynamics are crucial factors for the physiological functioning of neuronal communication. KEY POINTS: We developed a new probabilistic model of spontaneous and evoked vesicle fusion based on simple biophysical assumptions, including the motion of vesicles before they dock to the release site. We provide closed-form equations for the interval distribution of spontaneous releases in the special case of Brownian diffusion of vesicles, showing that a power-law heavy tail is generated. Fractional Brownian motion (fBm) was exploited to simulate anomalous vesicle diffusion, including directed and non-directed motion, by varying the Hurst exponent. We show that our model predicts non-linear features observed in experimental spontaneous quantal release series as well as ultrastructural data of synaptic vesicles spatial distribution. Evoked exocytosis based on a diffusion-replenished readily releasable pool might explain the emergence of power-law behaviour in neuronal activity.

Distinguishing subsampled power laws from other heavy-tailed distributions.

Distinguishing power-law distributions from other heavy-tailed distributions is challenging, and this task is often further complicated by subsampling effects. In this work, we evaluate the performance of two commonly used methods for detecting power-law distributions-the maximum likelihood method of Clauset etal. and the extreme value method of Voitalov etal.-in distinguishing subsampled power laws from two other heavy-tailed distributions, the lognormal and the stretched exponential distributions. We focus on a random subsampling method commonly applied in network science and biological sciences. In this subsampling scheme, we are ultimately interested in the frequency distribution of elements with a certain number of constituent parts-for example, species with k individuals or nodes with k connections-and each part is selected to the subsample with an equal probability. We investigate how well the results obtained from low-subsampling-depth subsamples generalize to the original distribution. Our results show that the power-law exponent of the original distribution can be estimated fairly accurately from subsamples, but classifying the distribution correctly is more challenging. The maximum likelihood method falsely rejects the power-law hypothesis for a large fraction of subsamples from power-law distributions. While the extreme value method correctly recognizes subsampled power-law distributions with all tested subsampling depths, its capacity to distinguish power laws from the heavy-tailed alternatives is limited. However, these false positives tend to result not from the subsampling itself but from the estimators' inability to classify the original sample correctly. In fact, we show that the extreme value method can sometimes be expected to perform better on subsamples than on the original samples from the lognormal and the stretched exponential distributions, while the contrary is true for the main tests included in the maximum likelihood method.

Measuring income inequality via percentile relativities

The adage “the rich are getting richer” refers to increasingly skewed and heavily-tailed income distributions. For such distributions, the mean is not the best measure of the center, but the classical indices of income inequality, including the celebrated Gini index, are mean based. In view of this, it has been proposed in the literature to incorporate the median into the definition of the Gini index. In the present paper we make a further step in this direction and, to acknowledge the possibility of differing viewpoints, investigate three median-based indices of inequality. These indices overcome past limitations, such as: (1) they do not rely on the mean as the center of, or a reference point for, income distributions, which are skewed, and are getting even more heavily skewed; (2) they are suitable for populations of any degree of tail heaviness, and income distributions are becoming increasingly such; and (3) they are unchanged by, and even discourage, transfers among the rich persons, but they encourage transfers from the rich to the poor, as well as among the poor to alleviate their hardship. We study these indices analytically and numerically using various income distribution models. Real-world applications are showcased using capital incomes from 2001 and 2018 surveys from fifteen European countries.

An evaluation of variable selection methods using Southern Africa solar irradiation data

Dimensionality poses a challenge in developing quality predictive models. Often when modelling solar irradiance (SI), many covariates are considered. Training such data has several disadvantages. This study sought to identify the best variable embedded selection method for different location and time horizon combinations from Southern Africa solar irradiance data. It introduced new variable selection methods into solar irradiation studies, namely penalised quantile regression (PQR), regularised random forests (RRF), and quantile regression forest (QRF). Stability analysis, performance and accuracy metric evaluations were used to compare them with the common lasso, elastic and ridge regression methods. The QRF model performed best in all locations followed by the shrinkage methods on hourly data. However, it was found that QRF is not sensitive to associations through correlations, thereby ignoring the relevance of variables while focusing on importance. Among the shrinkage methods, the lasso performed best in only one location. On the 24-hour horizon, elastic net dominated the performances among the shrinkage methods, but QRF was best in three locations of the six considered. Results confirmed that variable selection methods performed differently on different situational data sets. Depending on the strengths of the methods, results were combined to identify the most paramount variables. Day, total rainfall, and wind direction were superfluous features in all situations. The study concluded that shrinkage methods are best in cases of extreme multicollinearity, while QRF is best on data sets with outliers or/and heavy tails.

Probability Distributions for Modeling Stock Market Returns—An Empirical Inquiry

Investing in stocks and shares is a common strategy to pursue potential gains while considering future financial needs, such as retirement and children’s education. Effectively managing investment risk requires thoroughly analyzing stock market returns and making informed predictions. Traditional models often utilize normal variance distributions to describe these returns. However, stock market returns often deviate from normality, exhibiting skewness, higher kurtosis, heavier tails, and a more pronounced center. This paper investigates the Laplace distribution and its generalized forms, including asymmetric Laplace, skewed Laplace, and the Kumaraswamy Laplace distribution, for modeling stock market returns. Our analysis involves a comparative study with the widely-used Variance-Gamma distribution, assessing their fit with the weekly returns of the S&P 500 Index and its eleven business sectors, drawing parallel inferences from international stock market indices like IBOVESPA and KOSPI for emerging and developed economies, as well as the 20+ Years Treasury Bond ETFs and individual stocks across varied time horizons. The empirical findings indicate the superior performance of the Kumaraswamy Laplace distribution, which establishes it as a robust alternative for precise return predictions and efficient risk mitigation in investments.

Scale Mixture of Gleser Distribution with an Application to Insurance Data

In this paper, the scale mixture of the Gleser (SMG) distribution is introduced. This new distribution is the product of a scale mixture between the Gleser (G) distribution and the Beta(a,1) distribution. The SMG distribution is an alternative to distributions with two parameters and a heavy right tail. We study its representation and some basic properties, maximum likelihood inference, and Fisher’s information matrix. We present an application to a real dataset in which the SMG distribution shows a better fit than two other known distributions.

Interval evaluation of wind power prediction based on skew-slash t distribution

Wind resources have characteristics of stochastic volatility, resulting in wind power forecast errors greatly. The short and medium-term wind power forecast error distribution of the actual wind farm operating system is analyzed, and a sharp peak and thick tail, with heavy tail and skewness characteristics, are presented. A skew-slash t distribution (SST distribution) is proposed to describe the error distribution of short-term wind power prediction, and the parameter estimation is carried out by probability density curve fitting. The fitting results of the wind power forecast error historical data of the Wind power plant based on SST distribution, t distribution, and t Location-slash distribution (TLS distribution) are obtained, and the fitting goodness and prediction accuracy are compared. The results show that the SST distribution can more effectively estimate the wind power prediction interval.

Robust 1-norm Periodograms for Analysis of Noisy Non-Gaussian Time Series with Irregular Cadences: Application to VLBI Astrometry of Quasars

Astronomical time series often have non-uniform sampling in time, or irregular cadences, with long gaps separating clusters of observations. Some of these data sets are also explicitly non-Gaussian with respect to the expected model fit, or the simple mean. The standard Lomb–Scargle periodogram is based on the least squares solution for a set of test periods and, therefore, is easily corrupted by a subset of statistical outliers or an intrinsically non-Gaussian population. It can produce completely misleading results for heavy-tailed distribution of residuals. We propose a robust 1-norm periodogram technique, which is based on the principles of robust statistical estimation. This technique can be implemented in weighted or unweighted options. The method is described in detail and compared with the classical least squares periodogram on a set of astrometric VLBI measurements of the ICRF quasar IERS B0642+449. It is uniformly applied to a collection of 259 ICRF3 quasars each with more than 200 epoch VLBI measurements, resulting in a list of 49 objects with quasi-periodic position changes above the 3σ level, which warrant further investigation.

Multifractal fluctuations in zebrafish (Danio rerio) polarization time series.

In this work, we study the polarization time series obtained from experimental observation of a group of zebrafish (Danio rerio) confined in a circular tank. The complex dynamics of the individual trajectory evolution lead to the appearance of multiple characteristic scales. Employing the Multifractal Detrended Fluctuation Analysis (MF-DFA), we found distinct behaviors according to the parameters used. The polarization time series are multifractal at low fish densities and their average scales with . On the other hand, they tend to be monofractal, and their average scales with for high fish densities. These two regimes overlap at critical density , suggesting the existence of a phase transition separating them. We also observed that for low densities, the polarization velocity shows a non-Gaussian behavior with heavy tails associated with long-range correlation and becomes Gaussian for high densities, presenting an uncorrelated regime.

Regularizing stock return covariance matrices via multiple testing of correlations

This paper develops a large-scale inference approach for the regularization of stock return covariance matrices. The framework allows for the presence of heavy tails and multivariate GARCH-type effects of unknown form among the stock returns. The approach involves simultaneous testing of all pairwise correlations, followed by setting non-statistically significant elements to zero. This adaptive thresholding is achieved through sign-based Monte Carlo resampling within multiple testing procedures, controlling either the traditional familywise error rate, a generalized familywise error rate, or the false discovery proportion. Subsequent shrinkage ensures that the final covariance matrix estimate is positive definite and well-conditioned while preserving the achieved sparsity. Compared to alternative estimators, this new regularization method demonstrates strong performance in simulation experiments and real portfolio optimization.

Modelling and Clustering Sea Conditions: Bivariate FiniteMixtures of Generalized Additive Models for Location, Shape, and Scale Applied to the Analysis of Meteorological Tides and Wave Heights

Modelling sea conditions is a complex task that requires a comprehensive analysis, considering various influencing factors. Observed and unobserved factors jointly play a role in the definition of sea conditions. Here, we consider finite mixtures of generalized linear additive models for location scale, and shape (GAMLSSs) to capture the effects of both environmental variables and omitted variables, whose effects are summarized using latent variables. The GAMLSS approach is flexible enough to allow for different data features such as non-normality, skewness, heavy tails, etc., and for the definition of a regression model not only for the expected values of the observed process but also for all the other distribution parameters, e.g., the variance. We collected data on multiple sea-related and environmental variables in Ancona (Italy) from two Italian networks: the Sea Level Measurement Network (Rete Mareografica Nazionale, RMN) and the Sea Waves Measurement Network (Rete Ondametrica Nazionale, RON). Our main outcomes were the meteorological tides (often also referred to as “residuals”) and the significant wave height. Atmospheric pressure and wind speed were considered as main drivers of the sea conditions, as well as the fetch associated with wind direction, linking these variables to the outcomes through the definition of multiple linear predictors in a regression framework. Our results confirm the importance of accounting for environmental variables and reveal that their effect is heterogeneous, where heterogeneity is modelled by three distinct mixture components, each capturing different sea conditions. These findings contribute to a deeper understanding of sea state dynamics and provide evidence of a clustering structure characterizing different sea conditions.

A robust beamforming with antenna selection approach for dense antenna arrays under limited CSI

This paper introduces a novel approach for dense antenna arrays, which integrates beamforming and antenna selection. The proposed method aims to optimize the number of transmit antennas to reduce power consumption and enhance spectral efficiency at the transmitter. By employing optimal beamforming weights, our approach steers signals toward the intended receiver to maximize the transmit power of the desired signal while mitigating side-lobe interference under channel uncertainty. To adapt beamforming parameters dynamically to changing channel conditions, we develop a channel estimation approach that obtains accurate channel state information (CSI). In contrast to prior studies, our channel estimation method utilizes a generalized Gaussian distribution (GGD) model, capable of accommodating both matched and mismatched CSI features by modeling heavy-tailed and light-tailed distributions. This allows our method to capture diverse channel statistical behaviors resulting from phenomena such as fading and interference under various propagation conditions. Furthermore, we introduce an angle entropy metric, which measures angle of departure (AoD) variations, indicating the degree of uncertainty regarding the directionality of transmitted signals. Simulation results validate the accuracy of our analytical findings and demonstrate the superior performance of the proposed approach in terms of robustness and bit-error-rate (BER) performance.