Pareto Distribution Research Articles

On Dagum-Pareto mixture distribution: properties, simulation and application in insurance

The most principal subject in actuarial science is the modeling of claims amount. There are two types of distributions associated with two types of risk. We speak of light-tailed distributions as opposed to heavy-tailed distributions or extreme value distribution. Among the distributions of the amount of heavy tail claim is: Weibull, Log-Normal, Pareto and Burr “Also named DAGUM”. Many attempts were implemented in expanding the classes of mixed and compound distributions. In this article, we present a new distribution by mixing Dagum distribution (α, λ, r) and Pareto distribution (α, λ) named Dagum-Pareto mixture distribution. Some of statistical properties of this new distribution are obtained which include reliability function, hazard function, skewness, kurtosis and quantile function. Also we use the maximum likelihood method to estimate the model parameters. A Simulation study was carried out to examine the performance and accuracy of the maximum likelihood estimates of the proposed distribution. An application of the new model to no-life insurance real data set is presented to illustrate its usefulness and applicability. In our case, it is proven that Dagum-Pareto mixture distribution provides better fit compared to others existing probability distributions using some goodness of fit measures.

The adaptive mean estimation of heavy tailed distribution under random censoring

We propose an alternative estimator of the heavy tailed mean under censored data relies with the threshold t which divides it into two parts for x ≤ t and x > t respectively. In the first parts we present an empirical distribution function under random censoring (rc). Then, we check the convergence of Lyapunov condition to ensuring the asymptotic normality of this parts. For the second parts we use the Peaks-Over-Threshold (POT) method of the tail distribution and fitting to the generalized Pareto distribution (GPD). Intentionally to give an alternative estimator of the tail expected value for the heavy tailed distribution when x > t under random censoring relies the KIB estimators of the GPD parameters. Ditto, we ensuring the asymptotic normality property of the KIB estimators with bias reduced and we present the asymptotic normality of the second parts when x > t. An asymptotically normally distributed estimate for the heavy tailed mean estimation with infinite variance under censored data is introduced by combining the two parts. Finally, numerical examples are presented at the end of the paper to demonstrate the reliability of this estimation procedure and to better illustrate the results of this paper.

Estimation of Bonferroni Curve and Bonferroni Index of the Pareto Distribution

Abstract In this article, we consider classical and Bayesian estimations of some economic measures, specially Bonferroni Curve and Bonferroni Index of the Pareto distribution. We obtain the Maximum Likelihood Estimator and Uniform Minimum Variance Unbiased Estimator in the classical setup, and their properties are studied. Additionally, we conduct Bayesian estimation procedures based on symmetric loss functions and a truncated gamma prior. The precision of the estimators is evaluated under different sample sizes via Monte Carlo simulation. Furthermore, a real dataset is provided to compute all the estimators.

Assessment of reliability and availability of a system by using a bivariate stochastic model

AbstractIn the present age of industrial revolution where the smart manufacturing and smart production are being used in the developed countries, the production systems are still vulnerable to disruptions and outages that affect the reliability and availability of the systems. Certain scientific and probabilistic approaches are required to study the inter‐failure times and repair times and find the possible solutions for maximum production and minimum losses. In this article, we propose a bivariate Power Pareto distribution to model the inter‐failure times and repair times of a system along with some of its characteristics. The model parameters are estimated by employing the maximum likelihood estimation, Bayesian estimation and ant colony optimization algorithm. A simulation study is conducted for different sample sizes to assess the stability of the model parameters. Moreover, we derive the distribution of convolution from our proposed bivariate stochastic distribution and compute its quantiles that help to determine the total time of availability and recovery of a certain system. To demonstrate the efficacy of our model, we use real data set of inter‐failure times and repair times of a certain system.

Retraction Note: A novel machine learning approach for software reliability growth modelling with pareto distribution function

Retraction Note: A novel machine learning approach for software reliability growth modelling with pareto distribution function

Impacts of climate and land-use change on flood events with different return periods in a mountainous watershed of North China

Study regionLuanhe River Basin in North China Study focusGlobal warming exacerbates hydrological cycles, leading to increased intensity and frequency of floods. Regional studies on the impacts of climate change and land cover changes on flood events in recent decades are particularly important for flood prevention and risk assessment. In this study, the Peak Over Threshold- Generalized Pareto Distribution (POT-GPD) method is used to fit extreme runoff values in the Luanhe River Basin from 1961 to 2018 to identify floods with different return periods. Six flood behavior indicators are adopted to analyze the flood characteristics and extreme climate indices are also used for breakpoint testing and trend analysis. Various scenarios are established as the input of the SWAT model to study the impacts of climate change and land-use change on the flood events. New hydrological insights for the regionThe results indicate that the 2-year and 20-year return period floods are primarily rapid-rise single-peak floods, dominated by short-duration intense precipitation. The 5-year and 10-year return period floods are mainly multi-peak floods, primarily caused by long-duration precipitation. Floods in the Luanhe River Basin are primarily influenced by climate change, however, flood events do not directly correspond to the return periods of precipitation events. Forest is the most effective land-use type for flood retention. It could reduce flood magnitudes, especially for the 20-year return period floods. However, increasing forest cover does not effectively reduce the occurrence of the 5-year or 10-year floods, i.e. multi-peak floods. With more extreme rainfall events triggered by future climate change, particular attention should be paid to long-duration precipitation events, which may result in more severe disasters and economic losses.

Statistical Inference for Hüsler–Reiss Graphical Models Through Matrix Completions

The severity of multivariate extreme events is driven by the dependence between the largest marginal observations. The Hüsler–Reiss distribution is a versatile model for this extremal dependence, and it is usually parameterized by a variogram matrix. In order to represent conditional independence relations and obtain sparse parameterizations, we introduce the novel Hüsler–Reiss precision matrix. Similarly to the Gaussian case, this matrix appears naturally in density representations of the Hüsler–Reiss Pareto distribution and encodes the extremal graphical structure through its zero pattern. For a given, arbitrary graph we prove the existence and uniqueness of the completion of a partially specified Hüsler–Reiss variogram matrix so that its precision matrix has zeros on non-edges in the graph. Using suitable estimators for the parameters on the edges, our theory provides the first consistent estimator of graph structured Hüsler–Reiss distributions. If the graph is unknown, our method can be combined with recent structure learning algorithms to jointly infer the graph and the corresponding parameter matrix. Based on our methodology, we propose new tools for statistical inference of sparse Hüsler–Reiss models and illustrate them on large flight delay data in the United States, as well as Danube river flow data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Optimization of distribution function and model parameters for molecular communication via diffusion with OtoO approximation

The analysis is generally conducted in stationary receiver and transmitter models in a diffusion environment for the fundamental Molecular communication (MOC) models. However, a mobile MOC model is employed in this study, deviating from the existing literature. This mobile MOC model considers the mobility of all variables in the diffusion environment, including the transmitter, receiver, and molecules. Firstly, a novel MOC model is proposed, departing from the conventional normal distribution for the mobility of variables. Instead, alternative distribution functions such as the Pareto distribution, extreme value distribution, t-distribution, and generalized extreme value distribution are employed. Furthermore, the system's performance is enhanced by optimizing the distribution function and model parameters, such as the diffusion coefficient, using the optimization of optimization (OtoO) approach. In this approach, the Multi-Verse Optimization (MVO) algorithm serves as the primary algorithm, while the Grey Wolf Optimization (GWO) algorithm functions as the auxiliary algorithm. Essentially, the MVO algorithm optimizes the parameters of the MOC model, while simultaneously, the GWO algorithm optimizes the impact of the optimization processes of MVO on the parameters ``p'' and ``N'' as well as the constant parameter of the distribution function. By optimizing both the parameters of the MOC model and the distribution function, the number of received molecules is significantly increased. Therefore, this study not only improves the results of the MOC model structure based on different distribution functions but also optimizes all parameters of the proposed model using the MVO-GWO OtoO approach.

Comparison of Various Ranking Methods with Pareto Distribution Methods in Imprecise Data

The Pareto distribution has seen wide application in many different fields recently. Nowadays, the data we collect may be imprecise and contain uncertainties. This means that the values we gather cannot necessarily be considered a single, definitive value. In this paper, it is explored how various estimation methods like the method of moments, maximum likelihood estimation, and least squares estimation are used to determine the parameters of a Pareto distribution when the data is fuzzy or uncertain in nature. When making decisions, ranking methods are often heavily relied upon. For this reason, the application of several different ranking techniques to triangular fuzzy numbers (TFNs) is considered. The ranking methods for TFNs are found to tend to produce estimated parameters that are quite similar or even identical. The estimating parameters of a Pareto distribution can be solved for when dealing with fuzziness by the use of ranking methods such as alpha-cut, Yeager’s method, sub-interval method, Pascal method, magnitude method, and centroid method on the TFNs. By comparing the results of these various ranking techniques, an analysis is conducted to whether the estimated parameters of the Pareto distribution are consistent across the different ranking methods for TFNs get the uniqueness of the solution or differ. The decision making done by among the various ranking techniques, which method preferable for future comparison with extended of new ranking techniques. The conclusion is that the simulation results match up well with actual application data, serving as a good example for explaining this overall strategy.

How to define the threshold of takeover response ability of different drivers in conditional automated driving

In conditional automated driving, the takeover response ability threshold is necessary for driver qualification assessment and liability division of automated vehicle accidents. The primary objective of this study is to establish a clear and quantifiable threshold for drivers’ takeover response ability in conditional automated driving scenarios. This threshold aims to serve as a benchmark for evaluating drivers’ readiness and developing safety regulations in automated driving. We designed 18 takeover events and invited 42 drivers to participate in the driving simulation experiment, and obtained their takeover time data. First, we analyze the differences of takeover time among drivers with different attributes (gender, age, driving year). Second, based on the Peaks Over Threshold and the generalized Pareto distribution model, we use the graphic method to calculate the range of takeover time threshold for drivers with different attributes. The result shows that the difference in the threshold range of takeover time between male and female drivers is relatively tiny. There are differences in the threshold range of takeover time for different age drivers, and the threshold is negatively correlated with age. Drivers with high driving experience within a safe range are allowed to have longer takeover times. Finally, the rationality of the takeover time threshold for drivers with different attributes has been verified. The return level curves are approximately linear (R2 > 0.77), indicating that the GPD model can capture the overall trend of the return level, which is changing with the probability level. This proves that the takeover time threshold is reasonable. This study uses TTCmin to calibrate takeover safety, and the takeover time threshold has a good classification performance for takeover safety (accuracy > 85 %). The above content proves the rationality of the takeover time threshold. The contribution of this study is to calculate the takeover time threshold of drivers with different attributes, which can help regulatory authorities assess the driver’s takeover response ability and support the liability division of automated vehicle accidents.

Calculating Premiums for Extreme-Tail Risks with Liability Claims and Nuclear Verdicts

Purpose: When insurance claims, particularly liability claims and nuclear verdicts, are governed by fat-tailed distributions, considerable uncertainty exists about the value of the tail index. Material and Methods: Using the theory of risk aversion, this paper establishes a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) and investigates its properties. Findings: Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims. This structure provides a natural model for incorporating tail-index uncertainty into premiums. The theory is illustrated through possible ‘premiums’ for liability claims and nuclear verdicts, utilizing the consistent pattern of observed extremes. Implications to Theory, Practice and Policy: By integrating statistical methods for tail index estimation and addressing the inherent uncertainty, the power principle offers a robust framework for determining premiums in high-risk environments. The paper concludes with practical implications for insurers, highlighting the need for advanced risk management strategies and regulatory considerations in dealing with extreme liability claims.

Continental Scale Regional Flood Frequency Analysis: Combining Enhanced Datasets and a Bayesian Framework

Flood frequency analysis at large scales, essential for the development of flood risk maps, is hindered by the scarcity of gauge flow data. Suitable methods are thus required to predict flooding in ungauged basins, a notoriously complex problem in hydrology. We develop a Bayesian hierarchical model (BHM) based on the generalized extreme value (GEV) and the generalized Pareto distribution for regional flood frequency analysis at high resolution across a large part of North America. Our model leverages annual maximum flow data from ≈20,000 gauged stations and a dataset of 130 static catchment-specific covariates to predict extreme flows at all catchments over the continent as well as their associated statistical uncertainty. Additionally, a modification is made to the data layer of the BHM to include peaks over threshold flow data when available, which improves the precision of the discharge level estimates. We validated the model using a hold-out approach and found that its predictive power is very good for the GEV distribution location and scale parameters and improvable for the shape parameter, which is notoriously hard to estimate. The resulting discharge return levels yield a satisfying agreement when compared with the available design peak discharge from various government sources. The assessment of the covariates’ contributions to the model is also informative with regard to the most relevant underlying factors influencing flood-inducing peak flows. According to the developed aggregate importance score, the key covariates in our model are temperature-related bioindicators, the catchment drainage area and the geographical location.

Anthropogenic and meteorological effects on the counts and sizes of moderate and extreme wildfires

AbstractThe growing frequency and size of wildfires across the US necessitates accurate quantitative assessment of evolving wildfire behavior to predict risk from future extreme wildfires. We build a joint model of wildfire counts and burned areas, regressing key model parameters on climate and demographic covariates. We use extended generalized Pareto distributions to model the full distribution of burned areas, capturing both moderate and extreme sizes, while leveraging extreme value theory to focus particularly on the right tail. We model wildfire counts with a zero‐inflated negative binomial model, and join the wildfire counts and burned areas sub‐models using a temporally‐varying shared random effect. Our model successfully captures the trends of wildfire counts and burned areas. By investigating the predictive power of different sets of covariates, we find that fire indices are better predictors of wildfire burned area behavior than individual climate covariates, whereas climate covariates are influential drivers of wildfire occurrence behavior.

The SPAR Model: A New Paradigm for Multivariate Extremes: Application to Joint Distributions of Metocean Variables

Abstract This paper presents the application of a new multivariate extreme value model for the estimation of metocean variables. The model requires fewer assumptions about the forms of the marginal distributions and dependence structure compared to existing approaches, and provides a flexible and rigorous framework for modeling multivariate extremes. The method involves a transformation of variables to polar coordinates. The tail of the radial variable is then modeled using the generalized Pareto distribution, with parameters conditional on angle, providing a natural extension of univariate theory to multivariate problems. The resulting model is referred to as the semi-parametric angular-radial (SPAR) model. We consider the estimation of the joint distributions of (1) wave height and wave period, and (2) wave height and wind speed. We show that the SPAR model provides a good fit to the observations in terms of both the marginal distributions and dependence structures. The use of the SPAR model for estimating long-term extreme responses of offshore structures is discussed, using some simple response functions for floating structures and an offshore wind turbine with monopile foundation. We show that the SPAR model is able to accurately reproduce response distributions, and provides a realistic quantification of uncertainty.

Degree distributions in networks: Beyond the power law

AbstractThe power law is useful in describing count phenomena such as network degrees and word frequencies. With a single parameter, it captures the main feature that the frequencies are linear on the log‐log scale. Nevertheless, there have been criticisms of the power law, for example, that a threshold needs to be preselected without its uncertainty quantified, that the power law is simply inadequate, and that subsequent hypothesis tests are required to determine whether the data could have come from the power law. We propose a modeling framework that combines two different generalizations of the power law, namely the generalized Pareto distribution and the Zipf‐polylog distribution, to resolve these issues. The proposed mixture distributions are shown to fit the data well and quantify the threshold uncertainty in a natural way. A model selection step embedded in the Bayesian inference algorithm further answers the question whether the power law is adequate.

Characterizing the distribution of extreme geoelectric field events in Sweden

Historically, Sweden has reported several impacts on transformers and transmission lines related to geomagnetically induced currents (GICs) that develop during strong space weather events. GICs are driven by the geoelectric field (E), and their intensity depends on various factors, including the lithology conductivity and the rate of change of the Earth's magnetic field. The purpose of this study is to perform an extreme value analysis (EVA) of the E magnitude at six different latitudes in Sweden and to express the maximum |E| that might be observed in 10, 50 and 100 years. We analyzed 10-second E data in Sweden, which was obtained from a 1-D model. This model incorporates 10-second geomagnetic measurements from the IMAGE network and the vertical Earth's ground electrical conductivity in Sweden, extracted from a 3-D conductance map for the Fennoscandian region. Extreme E events tend to occur in clusters around geomagnetic disturbances (substorms and geomagnetic storms). Therefore, we applied two different methods to decluster the data. After declustering, Generalized Pareto (GP) distributions were fitted to the remaining extreme events that exceed the 99.5th percentile. The EVA indicates that the shape parameter of the GP distribution depends on latitude. This implies that at higher geographic latitudes (64.52-68.02°N) the distribution decreases faster toward zero than at lower latitudes (58.26-62.25°N). As a result the expected maximum |E| in 100-years in central Sweden ranges between 4.0-8.5 V/km, while at higher latitudes, it ranges between 2.0-2.5 V/km similar to the modeled geoelectric field values during the Halloween event in October 2003. In particular, around 60.50°N the distribution of extreme events exhibits the heaviest tail. When we additionally consider the effect of conductivity, the region of west Sweden around 60.50°N exhibits the largest expected maximum in 100-years with a value around 8.5 V/km. This is three times larger the maximum modeled |E| at that latitude.

Global application of a regional frequency analysis to extreme sea levels

Abstract. Coastal regions face increasing threats from rising sea levels and extreme weather events, highlighting the urgent need for accurate assessments of coastal flood risk. This study presents a novel approach to estimating global extreme sea level (ESL) exceedance probabilities using a regional frequency analysis (RFA) approach. The research combines observed and modelled hindcast data to produce a high-resolution (∼1 km) dataset of ESL exceedance probabilities, including wave setup, along the entire global coastline (excluding Antarctica). The methodology presented in this paper is an extension of the regional framework of Sweet et al. (2022), with innovations introduced to incorporate wave setup and apply the method globally. Water level records from tide gauges and a global reanalysis of tide and surge levels are integrated with a global ocean wave reanalysis. Subsequently, these data are regionalised, normalised, and aggregated and then fit with a generalised Pareto distribution. The regional distributions are downscaled to the local scale using the tidal range at every location along the global coastline obtained from a global tide model. The results show 8 cm of positive bias at the 1-in-10-year return level when compared to individual tide gauges. The RFA approach offers several advantages over traditional methods, particularly in regions with limited observational data. It overcomes the challenge of short and incomplete observational records by substituting long historical records with a collection of shorter but spatially distributed records. These spatially distributed data not only retain the volume of information but also address the issue of sparse tide gauge coverage in less populated areas and developing nations. The RFA process is illustrated using Cyclone Yasi (2011) as a case study, demonstrating how the approach can improve the characterisation of ESLs in regions prone to tropical cyclone activity. In conclusion, this study provides a valuable resource for quantifying the global coastal flood risk, offering an innovative global methodology that can contribute to preparing for – and mitigating against – coastal flooding.

Different Statistical Inference Algorithms for the New Pareto Distribution Based on Type-II Progressively Censored Competing Risk Data with Applications

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

Extreme value statistics of nerve transmission delay.

Delays in nerve transmission are an important topic in the field of neuroscience. Spike signals fired or received by the dendrites of a neuron travel from the axon to a presynaptic cell. The spike signal then triggers a chemical reaction at the synapse, wherein a presynaptic cell transfers neurotransmitters to the postsynaptic cell, regenerates electrical signals via a chemical reaction through ion channels, and transmits them to neighboring neurons. In the context of describing the complex physiological reaction process as a stochastic process, this study aimed to show that the distribution of the maximum time interval of spike signals follows extreme-order statistics. By considering the statistical variance in the time constant of the leaky Integrate-and-Fire model, a deterministic time evolution model for spike signals, we enabled randomness in the time interval of the spike signals. When the time constant follows an exponential distribution function, the time interval of the spike signal also follows an exponential distribution. In this case, our theory and simulations confirmed that the histogram of the maximum time interval follows the Gumbel distribution, one of the three forms of extreme-value statistics. We further confirmed that the histogram of the maximum time interval followed a Fréchet distribution when the time interval of the spike signal followed a Pareto distribution. These findings confirm that nerve transmission delay can be described using extreme value statistics and can therefore be used as a new indicator of transmission delay.

Improved of Multiobjective Fuzzy Probabilistic Solid Transportation Models in Transportation Systems

Transportation activities are stages in the product distribution system. The purpose of the transport system is to minimize the total cost. If there is more than one objective function and consider more than one type of vehicle, it is called a multiobjective solid transportation problem. In some cases, the parameter transportation model is under uncertainty. A probabilistic and fuzzy approach can be used. This research introduces a probabilistic fuzzy multiobjective solid transportation model where the source, destination and vehicle parameters follow the Pareto distribution. A triangular fuzzy number expresses the coefficient of the objective function. The obtained model is applied to the problem of the metal crate transportation system. There are two objective functions; the first is the objective function to minimize the total cost. The second is the objective function to minimize the total transportation time. Three types of vehicles are considered: HDL, Engkel and Wingbox. The result is that the total cost is Rp. 3836595, and the total time is 757.245 minutes or 13 hours.