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The insurance has often been involved to minimize financial losses. As the product providers, the insurance companies must effectively manage risks to prevent errors in risk measurement. The amount of risk or loss experienced by the policyholder refers to the claim amount. The Value at Risk (VaR) is commonly used to measure risk. The VaR is calculated from the probability function, which can be obtained by evaluating the distribution of claims data. Most claim frequencies are small, but occasionally, huge claims appear. Therefore, the appropriate distribution would be characterized by a heavy-tailed. Thus, this research aims to model and evaluate insurance claims data using exponential, Weibull, Pareto, and lognormal distributions to assess financial risk through VaR. The insurance claims data were collected from a single insurance company and include 1,326 claims. This research specifically examines variables such as gender, diabetic status, smoking status, the number of claims, and the level of confidence. The data were analysed using descriptive statistics, Maximum Likelihood Estimation for parameter estimation, and Goodness of Fit tests to determine the best-fitting distribution, along with VaR calculations based on the results. The suitability of the distribution model is assessed through the VaR and is analysed based on the appropriate distribution of insurance claims data. It is obtained that the Weibull and lognormal distributions appropriately model insurance claims data. The highest VaR is observed in the claim data for female non-diabetic smokers, with a level of confidence of 99.5%. The lowest VaR is obtained from the claim data for male diabetic non-smokers, with a level of confidence of 90%. This approach enhances the prediction of large potential losses for specific demographic groups, aiding more informed decision-making in premium pricing and risk management. The integration of heavy-tailed distributions in risk assessment, with a particular focus on demographic specificity, constitutes a substantial and novel contribution to this research.

Abstract The estimation of a certain threshold beyond which an extreme value distribution can be fitted to the tail of a data distribution remains one of the main issues in the theory of statistics of extremes. While standard Peak over Threshold (PoT) approaches determine this threshold graphically, we introduce in this paper a general framework which makes it possible for one to determine this threshold algorithmically by estimating it as a free parameter within a composite distribution. To see how this threshold point arises, we propose a general framework for generating three-component hybrid distributions which meets the need of data sets with right heavy-tail. The approach involves the combination of a distribution which can efficiently model the bulk of the data around the mean, with an heavy-tailed distribution meant to model the data observations in the tail while using another distribution as a link to connect the two. Some special examples of distributions resulting from the general framework are generated and studied. An estimation algorithm based on the maximum likelihood method is proposed for the estimation of the free parameters of the hybrid distributions. Application of the hybrid distributions to the S &P 500 index financial data set is also carried out.

Abstract In this paper we propose an algorithm for testing whether the independent observations come from finite-variance distribution. The preliminary knowledge about the data properties may be crucial for its further analysis and selection of the appropriate model. The idea of the testing procedure is based on the simple observation that the empirical cumulative even moment (ECEM) for data from finite-moments distribution tends to some constant whereas for data coming from heavy-tailed distribution, the ECEM exhibits irregular chaotic behavior. Based on this fact, in this paper we parameterize the regular/irregular behavior of the ECEM and construct a new test statistic. The efficiency of the testing procedure is verified for simulated data from three heavy-tailed distributions with possible finite and infinite variances. The effectiveness is analyzed for data represented in time domain. The simulation study is supported by analysis of real vibration signals from rotating machines. Here, the analyses are provided for data in both the time and time-frequency domains.

The estimation of a certain threshold beyond which an extreme value distribution can be fitted to the tail of a data distribution remains one of the main issues in the theory of statistics of extremes. While standard Peak over Threshold (PoT) approaches determine this threshold graphically, we introduce in this paper a general framework which makes it possible for one to determine this threshold algorithmically by estimating it as a free parameter within a composite distribution. To see how this threshold point arises, we propose a general framework for generating three-component hybrid distributions which meets the need of data sets with right heavy-tail. The approach involves the combination of a distribution which can efficiently model the bulk of the data around the mean, with an heavy-tailed distribution meant to model the data observations in the tail while using another distribution as a link to connect the two. Some special examples of distributions resulting from the general framework are generated and studied. An estimation algorithm based on the maximum likelihood method is proposed for the estimation of the free parameters of the hybrid distributions. Application of the hybrid distributions to the S &P 500 index financial data set is also carried out. [ABSTRACT FROM AUTHOR]

In this paper we propose an algorithm for testing whether the independent observations come from finite-variance distribution. The preliminary knowledge about the data properties may be crucial for its further analysis and selection of the appropriate model. The idea of the testing procedure is based on the simple observation that the empirical cumulative even moment (ECEM) for data from finite-moments distribution tends to some constant whereas for data coming from heavy-tailed distribution, the ECEM exhibits irregular chaotic behavior. Based on this fact, in this paper we parameterize the regular/irregular behavior of the ECEM and construct a new test statistic. The efficiency of the testing procedure is verified for simulated data from three heavy-tailed distributions with possible finite and infinite variances. The effectiveness is analyzed for data represented in time domain. The simulation study is supported by analysis of real vibration signals from rotating machines. Here, the analyses are provided for data in both the time and time-frequency domains. [ABSTRACT FROM AUTHOR]

Distributed state estimation is one of the critical technologies in the field of target tracking, where the process noise and measurement noise may have a heavy-tailed distribution. Traditionally, heavy-tailed distributions like the student-t distribution are employed, but our observation reveals that Gaussian noise predominates in many instances, with occasional outliers. This sporadic reliance on heavy-tailed distributions can degrade performances or necessitate frequent parameter adjustments. To overcome this, we introduce a novel distributed consensus multi-distribution state estimation method that combines Gaussian and student-t filters. Our approach establishes a system model using both Gaussian and student-t distributions. We derive a multi-distribution filter for a single sensor, assigning probabilities to Gaussian and student-t noise models. Parallel estimation under both distributions, utilizing Gaussian and student-t filters, allows us to calculate the likelihood of each distribution. The fusion of these results yields a mixed-state estimation and corresponding error matrix. Recognizing the increasing degrees of freedom in the student-t distribution over time, we provide an effective approximation. An information consensus strategy for multi-distribution filters is introduced, achieving global estimation through consensus on fused local filter results via interaction with neighboring nodes. This methodology is extended to the distributed case, and the recursive process of the distributed multi-distribution consensus state estimation method is presented. Simulation results demonstrate that the estimation accuracy of the proposed algorithm improved by at least 20% compared to that of the traditional algorithm in scenarios involving both Gaussian and heavy-tailed distributions. [ABSTRACT FROM AUTHOR]

This article deals with the asymptotic distribution for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to − ∞ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider goodness-of-fit methods for multivariate symmetric and asymmetric stable Paretian random vectors in arbitrary dimension. The methods are based on the empirical characteristic function and are implemented both in the i.i.d. context as well as for innovations in GARCH models. Asymptotic properties of the proposed procedures are discussed, while the finite-sample properties are illustrated by means of an extensive Monte Carlo study. The procedures are also applied to real data from the financial markets. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Modelling the volatility of commodity prices and creating more reliable models for estimating and forecasting commodity price returns are crucial. The body of research on statistical models that can fully reflect the empirical characteristics of commodity price returns is lacking. The main aim of this research was to develop a modelling framework that could be used to accurately estimate and forecast commodity price returns by combining long memory models with heavy-tailed distributions. This study employed dual hybrid long-memory generalised autoregressive conditionally heteroscedasticity (GARCH) models with heavy-tailed innovations, namely, the Student-t distribution (StD), skewed-Student-t distribution (SStD), and the generalised error distribution (GED). Based on the smallest forecasting metrics values for mean absolute error (MAE) and mean squared error (MSE) values, the best performing LM-GARCH-type model for lithium is the ARFIMA (1, o , 1)-FIAPARCH (1, ξ , 1) with normal innovations. For tobacco, the best model is ARFIMA (1, o , 1)-FIGARCH (1, ξ , 1) with SStD innovations. The robust performing model for gold is the ARFIMA (1, o , 1)-FIGARCH (1, ξ , 1)-GED model. The best performing forecasting model for crude oil and cotton returns are the F I A P A R C H   1 , ξ ,   1 − S S t D model and H Y G A R C H   1 , ξ ,   1 − S t D model, respectively. The results obtained from this study would be beneficial to those concerned with financial market modelling techniques, such as derivative pricing, risk management, asset allocation, and valuation. [ABSTRACT FROM AUTHOR]

In this article, we introduce a robust generalization of the generalized Topp-Leone-G (GEN-TL-G) family of distributions via the heavy-tailed technique, namely, heavy-tailed generalized Topp-Leone-G (HT-GEN-TL-G) family of distributions. Statistical properties of the HT-GEN-TL-G family of distributions including reliability functions, quantile function, density expansion, moments, moment generating function, incomplete moments, Rényi entropy, distribution of order statistics are derived. Different estimation methods including Maximum Likelihood, Anderson-Darling, Ordinary Least Squares, Weighted Least Squares, Cramér-von Mises and Maximum Product of Spacing are utilized to estimate the unknown parameters of the new distribution, and a simulation study is used to compare the results of the estimation methods. Risk measures for this distribution were also developed and finally the effectiveness of this new family of distributions was demonstrated using applications to two real data sets. Graphical and application results in this manuscript were obtained using R programming language. [ABSTRACT FROM AUTHOR]

Due to the direct statistical interpretation, censored linear regression offers a valuable complement to the Cox proportional hazards regression in survival analysis. We propose a rank-based high-dimensional inference for censored linear regression without imposing any moment condition on the model error. We develop a theory of the high-dimensional U-statistic, circumvent challenges stemming from the non-smoothness of the loss function, and establish the convergence rate of the regularized estimator and the asymptotic normality of the resulting de-biased estimator as well as the consistency of the asymptotic variance estimation. As censoring can be viewed as a way of trimming, it strengthens the robustness of the rank-based high-dimensional inference, particularly for the heavy-tailed model error or the outlier in the presence of the response. We evaluate the finite-sample performance of the proposed method via extensive simulation studies and demonstrate its utility by applying it to a subcohort study from The Cancer Genome Atlas (TCGA). [ABSTRACT FROM AUTHOR]

Studying the extreme value theory (EVT) involves multiple main objectives, among them the estimation of the tail index parameter. Some estimation methods are used to estimate the tail index parameter like maximum likelihood estimation (MLE). Additionally, the Hill estimator is one type of maximum likelihood estimator, which is a more robust with a large sample than a small sample. This research proposes the construction of an alternative estimator for the parameter of the heavy-tailed distribution using the maximum lq-likelihood estimation (MLqE) approach in order to adapt the ML and Hill estimator with the small sample. Furthermore, the maximum lq-likelihood estimator asymptotic normality is established. Moreover, several simulation studies in order to compare the MLq estimator with the ML estimators are provided. In the excesses over high suitable threshold values the number of the largest observation k will lead to an efficient estimate of the Hill estimator. For this, selection of k in the Hill estimator was investigated using the method of the quantile type 8 which is effective with the hydrology data. The performance of the Hill estimator and the lq-Hill estimator is subsequently compared by employing real relies with the distribution of hydrology data.

Using daily closing price data spreading over 3 April 1990, to 5 May 2020, this study explores the skewness and excess kurtosis behaviour across energy, metals, and agricultural commodity futures. Subsequently, it compares the fitting of an empirical distribution under normal distribution assumption with those under the Generalized Hyperbolic distribution. The generalized hyperbolic distribution includes Hyperbolic distribution, Variance-Gamma distribution, and Normal Inverse Gaussian distribution. The results show that the Normal Inverse Gaussian distribution for natural gas, gold, platinum, copper, sugar, and feeder cattle futures captures skewness as well as excess kurtosis of the daily logarithmic returns. The findings are robust to the sub-sample analysis. [ABSTRACT FROM AUTHOR]

We are occupied with the issue of consistency of the Gaussian QMLE in GARCH-type models with very heavy tailed squared innovations. We show that the appropriately scaled likelihood function weakly epi-converges to a stochastic process that is a.s. lower semi-continuous and proper. When moreover the volatility filter is increasing w.r.t. the parameter, inconsistency follows due to that the true parameter value misses the set of minimizers of the limit. This holds for models like the AGARCH, the Augmented GARCH, and the GQARCH. [ABSTRACT FROM AUTHOR]

We consider goodness‐of‐fit tests for the multivariate Student's t‐distribution with i.i.d. data and for the innovation distribution in a generalized autoregressive conditional heteroskedasticity model. The methods are based on the empirical characteristic function and are relatively easy to implement, invariant under linear transformations, and globally consistent. Asymptotic properties of the proposed procedures are investigated, while the finite‐sample properties are illustrated by means of a Monte Carlo study. The procedures are also applied to real data from the financial markets. [ABSTRACT FROM AUTHOR]

The λ-exponential family has recently been proposed to generalize the exponential family. While the exponential family is well-understood and widely used, this is not the case yet for the λ-exponential family. However, many applications require models that are more general than the exponential family, and the λ-exponential family is often a good alternative. In this work, we propose a theoretical and algorithmic framework to solve variational inference and maximum likelihood estimation problems over the λ-exponential family. We give new sufficient optimality conditions for variational inference problems. Our conditions take the form of generalized moment-matching conditions and generalize existing similar results for the exponential family. We exhibit novel characterizations of the solutions of maximum likelihood estimation problems, that recover optimality conditions in the case of the exponential family. For the resolution of both problems, we propose novel proximal-like algorithms that exploit the geometry underlying the λ-exponential family. These new theoretical and methodological insights are tested on numerical examples, showcasing their usefulness and interest, especially on heavy-tailed target distributions. [ABSTRACT FROM AUTHOR]

The class of α -stable distributions is widely used in various applications, especially for modeling heavy-tailed data. Although the α -stable distributions have been used in practice for many years, new methods for identification, testing, and estimation are still being refined and new approaches are being proposed. The constant development of new statistical methods is related to the low efficiency of existing algorithms, especially when the underlying sample is small or the distribution is close to Gaussian. In this paper, we propose a new estimation algorithm for the stability index, for samples from the symmetric α -stable distribution. The proposed approach is based on a quantile conditional variance ratio. We study the statistical properties of the proposed estimation procedure and show empirically that our methodology often outperforms other commonly used estimation algorithms. Moreover, we show that our statistic extracts unique sample characteristics that can be combined with other methods to refine existing methodologies via ensemble methods. Although our focus is set on the symmetric α -stable case, we demonstrate that the considered statistic is insensitive to the skewness parameter change, so our method could be also used in a more generic framework. For completeness, we also show how to apply our method to real data linked to financial market and plasma physics. [ABSTRACT FROM AUTHOR]

Prediction of loss reserves corresponding to dependent lines of business is one of the most important problems in the actuarial sciences. In this paper, we propose a class of copula based multivariate distributions to model the losses with the heavy tailed distribution in the run-off triangles to predict unpaid losses. We set up ANOVA, ANCOVA, and state space models with four choices of copulas, Clayton, Frank, Gumbel, and Gaussian to provide a new procedure for analyzing run-off triangle tables. We use the Hamiltonian Monte Carlo sampler to perform a Bayesian analysis to estimate the parameters. We apply the proposed models to the data set consists of two lines of business of paid losses data from the Schedule P of the National Association of Insurance Commissioners (NAIC) database. Using some well known criteria, we compare the prediction accuracy of the mean models. As a result, the ANCOVA model with the Clayton copula dominates the other models. [ABSTRACT FROM AUTHOR]

We propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log-quantiles and their neural network approximation is established. The finite sample performances of the non-conditional neural network estimator are compared to other bias-reduced extreme-value competitors on simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail. Finally, the conditional neural network estimators are implemented to investigate the behavior of extreme rainfalls as functions of their geographical location in the southern part of France. The source code is available at . [ABSTRACT FROM AUTHOR]

This paper studies the joint tail behavior of two randomly weighted sums ∑i=1m ΘiXi and ∑j=1nθjYj for some m, n ∈ ℕ ∪{∞}, in which the primary random variables {Xi;i ∈ ℕ} and {Yi;i ∈ ℕ}, respectively, are real-valued, dependent and heavy-tailed, while the random weights {Θi, θi; i ∈ ℕ} are nonnegative and arbitrarily dependent, but the three sequences {Xi;i ∈ ℕ}, {Yi;i ∈ ℕ} and {Θi, θi;i ∈ ℕ} are mutually independent. Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights, we establish some (uniformly) asymptotic formulas for the joint tail probability of the two randomly weighted sums, expressing the insensitivity with respect to the underlying weak dependence structures. As applications, we consider both discrete-time and continuous-time insurance risk models, and obtain some asymptotic results for ruin probabilities. [ABSTRACT FROM AUTHOR]

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum Sν = X1 + ··· + Xν of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of {X1,X2, . . .}. The conditions are provided for the relation P(Sν > x) ∼ Eν P(X1 > x) to hold, as x →∞, involving the finiteness of E|X1|. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment E|X1|r for some r > 1 was assumed. [ABSTRACT FROM AUTHOR]

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum ${S_{\nu }}={X_{1}}+\cdots +{X_{\nu }}$ of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of $\{{X_{1}},{X_{2}},\dots \}$. The conditions are provided for the relation $\mathbb{P}({S_{\nu }}\gt x)\sim \mathbb{E}\nu \hspace{0.1667em}\mathbb{P}({X_{1}}\gt x)$ to hold, as $x\to \infty $, involving the finiteness of $\mathbb{E}|{X_{1}}|$. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment $\mathbb{E}|{X_{1}}{|^{r}}$ for some $r\gt 1$ was assumed.

The classical Pareto distribution is a positively skewed and right heavy-tailed lifetime distribution having many applications in various fields of science and social science. In this work, via the logarithmic transformed method, a new three-parameter lifetime distribution, an extension of classical Pareto distribution is generated. The different structural properties of the new distribution are studied. The model parameters are estimated by the method of maximum likelihood and Bayesian procedure. When all the three parameters of the distribution are unknown, the Bayes estimators cannot be obtained in a closed form, and hence, the Lindley's approximation under squared error loss function is used to compute the Bayes estimators. A Monte Carlo simulation study is also conducted to compare the performance of these estimators using mean square error. The application of the new distribution for modeling earthquake insurance and reliability data are illustrated using two real data sets. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) dynamics have been the focus of epidemiological and biostatistical research during the past decades to understand the progression of acquired immunodeficiency syndrome (AIDS) in the population. Although there are several approaches for modeling HIV dynamics, one of the most popular is based on Gaussian mixed‐effects models because of its simplicity from the implementation and interpretation viewpoints. However, in some situations, Gaussian mixed‐effects models cannot (a) capture serial correlation existing in longitudinal data, (b) deal with missing observations properly, and (c) accommodate skewness and heavy tails frequently presented in patients' profiles. For those cases, mixed‐effects state‐space models (MESSM) become a powerful tool for modeling correlated observations, including HIV dynamics, because of their flexibility in modeling the unobserved states and the observations in a simple way. Consequently, our proposal considers an MESSM where the observations' error distribution is a skew‐t. This new approach is more flexible and can accommodate data sets exhibiting skewness and heavy tails. Under the Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is implemented. To evaluate the properties of the proposed models, we carried out some exciting simulation studies, including missing data in the generated data sets. Finally, we illustrate our approach with an application in the AIDS Clinical Trial Group Study 315 (ACTG‐315) clinical trial data set. [ABSTRACT FROM AUTHOR]

We consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric $\alpha$ -stable distribution, where $\alpha\in(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $\alpha$ -stable distribution having the same $\alpha$ parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $\alpha$ -stable distributions, $\alpha\in(0,2]$. [ABSTRACT FROM AUTHOR]

This paper considers a delayed claim risk model with constant interest rate when there is a dependence structure between the delayed claim and claim amount. We will incorporate the heavy-tailed distributions into the perturbed renewal risk model and obtain the uniform asymptotic estimate for the probability that an insurance portfolio gets ruined within a finite time period using some probability inequalities and mathematical approaches. Moreover, two numerical examples via Monte carlo simulation methods are presented to illustrate the effectiveness of results when the claim amount and delayed claim are dependent according to the Farlie–Gumbel–Morgenstern copula for Pareto and Lognormal distributions. [ABSTRACT FROM AUTHOR]

Transport engineers employ various interventions to enhance traffic-network performance. Quantifying the impacts of Cycle Superhighways is complicated due to the non-random assignment of such an intervention over the transport network. Treatment effects on asymmetric and heavy-tailed distributions are better reflected at extreme tails rather than at the median. We propose a novel method to estimate the treatment effect at extreme tails incorporating heavy-tailed features in the outcome distribution. The analysis of London transport data using the proposed method indicates that the extreme traffic flow increased substantially after Cycle Superhighways came into operation. [ABSTRACT FROM AUTHOR]

A renewal-reward process with dependent components and heavy-tailed interarrival times is investigated, and an asymptotic expansion as t - 8 for the expectation is derived. [ABSTRACT FROM AUTHOR]

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.

Stochastic Configuration Network (SCN) has a powerful capability for regression and classification analysis. Traditionally, it is quite challenging to correctly determine an appropriate architecture for a neural network so that the trained model can achieve excellent performance for both learning and generalization. Compared with the known randomized learning algorithms for single hidden layer feed-forward neural networks, such as Randomized Radial Basis Function (RBF) Networks and Random Vector Functional-link (RVFL), the SCN randomly assigns the input weights and biases of the hidden nodes in a supervisory mechanism. Since the parameters in the hidden layers are randomly generated in uniform distribution, hypothetically, there is optimal randomness. Heavy-tailed distribution has shown optimal randomness in an unknown environment for finding some targets. Therefore, in this research, the authors used heavy-tailed distributions to randomly initialize weights and biases to see if the new SCN models can achieve better performance than the original SCN. Heavy-tailed distributions, such as Lévy distribution, Cauchy distribution, and Weibull distribution, have been used. Since some mixed distributions show heavy-tailed properties, the mixed Gaussian and Laplace distributions were also studied in this research work. Experimental results showed improved performance for SCN with heavy-tailed distributions. For the regression model, SCN-Lévy, SCN-Mixture, SCN-Cauchy, and SCN-Weibull used less hidden nodes to achieve similar performance with SCN. For the classification model, SCN-Mixture, SCN-Lévy, and SCN-Cauchy have higher test accuracy of 91.5%, 91.7% and 92.4%, respectively. Both are higher than the test accuracy of the original SCN.

Distributed state estimation is one of the critical technologies in the field of target tracking, where the process noise and measurement noise may have a heavy-tailed distribution. Traditionally, heavy-tailed distributions like the student-t distribution are employed, but our observation reveals that Gaussian noise predominates in many instances, with occasional outliers. This sporadic reliance on heavy-tailed distributions can degrade performances or necessitate frequent parameter adjustments. To overcome this, we introduce a novel distributed consensus multi-distribution state estimation method that combines Gaussian and student-t filters. Our approach establishes a system model using both Gaussian and student-t distributions. We derive a multi-distribution filter for a single sensor, assigning probabilities to Gaussian and student-t noise models. Parallel estimation under both distributions, utilizing Gaussian and student-t filters, allows us to calculate the likelihood of each distribution. The fusion of these results yields a mixed-state estimation and corresponding error matrix. Recognizing the increasing degrees of freedom in the student-t distribution over time, we provide an effective approximation. An information consensus strategy for multi-distribution filters is introduced, achieving global estimation through consensus on fused local filter results via interaction with neighboring nodes. This methodology is extended to the distributed case, and the recursive process of the distributed multi-distribution consensus state estimation method is presented. Simulation results demonstrate that the estimation accuracy of the proposed algorithm improved by at least 20% compared to that of the traditional algorithm in scenarios involving both Gaussian and heavy-tailed distributions.

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.

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.

Modelling the volatility of commodity prices and creating more reliable models for estimating and forecasting commodity price returns are crucial. The body of research on statistical models that can fully reflect the empirical characteristics of commodity price returns is lacking. The main aim of this research was to develop a modelling framework that could be used to accurately estimate and forecast commodity price returns by combining long memory models with heavy-tailed distributions. This study employed dual hybrid long-memory generalised autoregressive conditionally heteroscedasticity (GARCH) models with heavy-tailed innovations, namely, the Student-t distribution (StD), skewed-Student-t distribution (SStD), and the generalised error distribution (GED). Based on the smallest forecasting metrics values for mean absolute error (MAE) and mean squared error (MSE) values, the best performing LM-GARCH-type model for lithium is the ARFIMA (1, o, 1)-FIAPARCH (1, ξ, 1) with normal innovations. For tobacco, the best model is ARFIMA (1, o, 1)-FIGARCH (1, ξ, 1) with SStD innovations. The robust performing model for gold is the ARFIMA (1, o, 1)-FIGARCH (1, ξ, 1)-GED model. The best performing forecasting model for crude oil and cotton returns are the FIAPARCH 1,ξ, 1−SStD model and HYGARCH 1,ξ, 1−StD model, respectively. The results obtained from this study would be beneficial to those concerned with financial market modelling techniques, such as derivative pricing, risk management, asset allocation, and valuation.

The adequate choice of a distribution that can fit a dataset, especially to its right tail (large extreme events), is a major problem in flood frequency analysis. Decision support systems (DSS) have been used in the past to define the appropriate class of distribution based on the tail behaviour of the data before its model selection. This paper investigates the tail behaviour of probability distribution of the daily streamflow data in south Indian rivers and also assesses the information related to tail risk, as it has many practical and societal consequences. In this paper, we apply and compare two DSS, (i) given by Martel et al. (J Hydrol Eng 18(1):1–9, 2013) and (ii) concentration profile–concentration adjusted expected shortfall (CP–CAES) based DSS, along with some newly developed graphical diagnostic tools, such as CP, CAES, discriminant moment ratio plot, maximum-to-sum plot, and Zenga plot to characterize the tails of probability distributions into an appropriate class. Further, the tail risk is analyzed using a novel risk management approach known as a concentration map (CM), which makes use of the concentration profiles of daily streamflow datasets. Results indicate that the proposed DSS is a potential tool for tail characterization. The study suggests the use of heavy-tailed distributions to model daily streamflow data over south Indian catchments. Neglecting heavy-tailed distributions, when found appropriate, can lead to an underestimation of the likelihood of floods and has catastrophic consequences for risk. CM is found suitable for assessing the tail risk associated with the daily streamflow dataset, which inherently represents the frequency and magnitude of extreme floods. [ABSTRACT FROM AUTHOR]

Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to the Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided. [ABSTRACT FROM AUTHOR]

Despite the growing interest in realized stochastic volatility models, their estimation techniques, such as simulated maximum likelihood (SML), are computationally intensive. Based on the realized volatility equation, this study demonstrates that, in a finite sample, the quasi-maximum likelihood estimator based on the Kalman filter is competitive with the two-step SML estimator, which is less efficient than the SML estimator. Regarding empirical results for the S&P 500 index, the quasi-likelihood ratio tests favored the two-factor realized asymmetric stochastic volatility model with the standardized t distribution among alternative specifications, and an analysis on out-of-sample forecasts prefers the realized stochastic volatility models, rejecting the model without the realized volatility measure. Furthermore, the forecasts of alternative RSV models are statistically equivalent for the data covering the global financial crisis. [ABSTRACT FROM AUTHOR]

Expected shortfall (ES), also known as superquantile or conditional value-at-risk, is an important measure in risk analysis and stochastic optimisation and has applications beyond these fields. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a joint regression framework recently proposed to model the quantile and ES of a response variable simultaneously, given a set of covariates. The current state-of-the-art approach to this problem involves minimising a non-differentiable and non-convex joint loss function, which poses numerical challenges and limits its applicability to large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity to nuisance parameters, we propose a statistically robust and computationally efficient two-step procedure for fitting joint quantile and ES regression models that can handle highly skewed and heavy-tailed data. We establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors that lay the foundation for statistical inference, even with increasing covariate dimensions. Finally, through numerical experiments and two data applications, we demonstrate that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression. [ABSTRACT FROM AUTHOR]

In this article, we study stochastic orders over an interval. Mainly, we focus on orders related to the Laplace transform. The results are then applied to obtain a bound for heavy-tailed distributions and are illustrated by some examples. We also indicate how these ordering relationships can be adapted to the classical risk model in order to derive a moment bound for ruin probability. Finally, we compare it with other existing bounds. [ABSTRACT FROM AUTHOR]

Consider a stable M/G/1 system in which, at time t = 0 , there are exactly n customers with residual service times equal to v 1 , v 2 , ... , v n . In addition, assume that there is an extra customer c who arrives at time t = 0 and has a service requirement of x. The externalities which are created by c are equal to the total waiting time that others will save if her service requirement is reduced to zero. In this work, we study the joint distribution (parameterized by n , v 1 , v 2 , ... , v n , x ) of the externalities created by c when the underlying service distribution is either last-come, first-served with preemption or first-come, first-served. We start by proving a decomposition of the externalities under the above-mentioned service disciplines. Then, this decomposition is used to derive several other results regarding the externalities: moments, asymptotic approximations as x → ∞ , asymptotics of the tail distribution, and a functional central limit theorem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of randomly weighted sums of the form of S n θ ξ = θ 1 ξ 1 + ⋯ + θ n ξ n under the transformation φ: ℝ → ℝ satisfying several asymptotic properties. The collection {ξ1,...,ξn} consists of dominatedly varying, not necessarily identically distributed, random variables following a specific dependence structure, whereas {θ1,...,θn} comprise of possibly dependent nonnegative and nondegenerate at zero random weights. Both collections are assumed to be independent. Inspired by the recent results. we obtain asymptotic bounds for the tail expectation E φ S n θ ξ 1 S n θ ξ > x expressing them by the sums of tail expectations E φ θ k ξ k 1 θ k ξ k > x . We provide several particular examples to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The proposed research incorporates the utilization of a heavy-tailed skewed distribution referred to as the inverse Weibull as a link function in the context of a binary classification model. This selection is motivated by the need to address the existence of rare or extreme events in random processes. The study introduces a model that relies on the Inverse Weibull (TYPE II) distribution, and the estimation of model parameters is accomplished through the application of maximum likelihood methods. When the outcomes are compared to those derived from other link functions such as TYPE I (Complementary log) and TYPE III (Weibull) based on extreme value distributions using standard classification data as well as real-life data, it becomes apparent that the Inverse Weibull (TYPE II) model exhibits exceptional performance. This assessment of performance takes into account several criteria, encompassing the Akaike information criterion, Bayesian information criterion, Area under the curve, and Brier scores. In conclusion, the study establishes that the proposed model demonstrates considerable robustness in its performance, rendering it a viable choice for the modeling of binary classification problems. [ABSTRACT FROM AUTHOR]

Stochastic Configuration Network (SCN) has a powerful capability for regression and classification analysis. Traditionally, it is quite challenging to correctly determine an appropriate architecture for a neural network so that the trained model can achieve excellent performance for both learning and generalization. Compared with the known randomized learning algorithms for single hidden layer feed-forward neural networks, such as Randomized Radial Basis Function (RBF) Networks and Random Vector Functional-link (RVFL), the SCN randomly assigns the input weights and biases of the hidden nodes in a supervisory mechanism. Since the parameters in the hidden layers are randomly generated in uniform distribution, hypothetically, there is optimal randomness. Heavy-tailed distribution has shown optimal randomness in an unknown environment for finding some targets. Therefore, in this research, the authors used heavy-tailed distributions to randomly initialize weights and biases to see if the new SCN models can achieve better performance than the original SCN. Heavy-tailed distributions, such as Lévy distribution, Cauchy distribution, and Weibull distribution, have been used. Since some mixed distributions show heavy-tailed properties, the mixed Gaussian and Laplace distributions were also studied in this research work. Experimental results showed improved performance for SCN with heavy-tailed distributions. For the regression model, SCN-Lévy, SCN-Mixture, SCN-Cauchy, and SCN-Weibull used less hidden nodes to achieve similar performance with SCN. For the classification model, SCN-Mixture, SCN-Lévy, and SCN-Cauchy have higher test accuracy of 91.5%, 91.7% and 92.4%, respectively. Both are higher than the test accuracy of the original SCN.

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy‐tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second‐order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset. [ABSTRACT FROM AUTHOR]