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Generalized Pareto distribution (GPD), an asymmetrical distribution, primarily models exceedances over a high threshold in many applications. Within the peaks-over-threshold (POT) framework, we consider a new GPD parameter estimation method to estimate a common tail risk measure, the value at risk (VaR). The proposed method is more suitable for the POT framework and makes full use of data information. Specifically, our estimation method builds upon the generalized probability weighted moments method and integrates it with the nonlinear weighted least squares method. We use exceedances for the GPD, minimizing the sum of squared differences between the sample and population moments of a function of GPD random variables. At the same time, the proposed estimator uses three iterations and assigns weight to further improving the estimated performance. Under Monte Carlo simulations and with a real heavy-tailed dataset, the simulation results show the advantage of the newly proposed estimator, particularly when VaRs are at high confidence levels. In addition, by simulating other heavy-tailed distributions, our method still exhibits good performance in estimating misjudgment distributions. [ABSTRACT FROM AUTHOR]

Estimating generalised Pareto distribution (GPD) parameters is a fundamental step in modelling the extremevalue distribution of random variables. It is generally done with the maximum likelihood method, but there are generally difficulties in estimating GPD parameters using this method as there is no closed-form solution for the first derivative of the GPD log-likelihood function. This makes the solution difficult to determine analytically. However, numerical methods can be used as an alternative. Therefore, this study estimates the solution numerically using a simple fixed-point iteration method that is intuitive for both practitioners and professionals. We obtained three fixed-point iterations when estimating GPD parameters that met the unbiased estimator and convergence criteria. The iterations allow practitioners and professionals to directly and efficiently estimate GPD parameters when modelling extreme-value distributions of random variables. [ABSTRACT FROM AUTHOR]

The Pareto distribution plays an important role in many data analysis tasks. An important aspect of this distribution is the estimation of its parameters. Several studies use classical methods, Bayes, and the neural network (NN) to evaluate Pareto parameters. Others have attempted to combine classical methods with a single NN-based. However, there isn't enough research to determine the sensitivity of the single NN to the specifics of the training data due to its stochastic training algorithm in the parameter estimation field. The current research aims to prove the efficiency of the aggregation of weighted multiple NN models and weighted ordinary least-squares regression algorithm to overcome the specifics of the training data and the sensitivity to outliers, respectively. The proposed method enables a locally less accurate model to participate to a lesser extent in the overall aggregation. The proposed method was compared with prevalent methods in the area, including the ordinary least squares, weighted ordinary least squares, maximum likelihood estimation, and the Bayes' using Monte Carlo simulations. The results verified the superiority of the proposed method in terms of regression error metrics. Moreover, it can be adapted to a variety of distributions. [ABSTRACT FROM AUTHOR]

In recent years, several probability distributions have been introduced in the literature to analyze the data exhibiting an upside-down bathtub–shaped failure rate; an inverse Pareto distribution (IPD) is an appropriate choice among them. For stress–strength reliability models, estimation of parameters is an interesting area of research. In this article, we estimate the stress–strength reliability parameter R = P (X > Y) (where X and Y are strength and stress variables, respectively) of the IPD, whereby we focus on the problem of fixed-accuracy confidence interval estimation of R. It is established that the proposed interval estimation problem cannot be solved with the help of any fixed sample technique. As a result, we propose a two-stage sequential sampling strategy (which reduces the sample size significantly) to solve the given estimation problem. We obtain the expressions of several exact operating characteristics associated with our two-stage sampling technique. We also establish that the proposed two-stage procedure enjoys interesting first-order asymptotic properties. The detailed simulation analyses support our theoretical findings, and two real data sets based on insurance claims reinforce the practical utility of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper conducts a systematic statistical analysis of the characteristics of the geographical empirical distributions for the numbers of both cumulative and daily confirmed COVID-19 cases and deaths at county, city, and state levels over a time span from January 2020 to June 2022. The mathematical heavy-tailed distributions can be used for fitting the empirical distributions observed in different temporal stages and geographical scales. The estimations of the shape parameter of the tail distributions using the Generalized Pareto Distribution also support the observations of the heavy-tailed distributions. According to the characteristics of the heavy-tailed distributions, the evolution course of the geographical empirical distributions can be divided into three distinct phases, namely the power-law phase, the lognormal phase I, and the lognormal phase II. These three phases could serve as an indicator of the severity degree of the COVID-19 pandemic within an area. The empirical results suggest important intrinsic dynamics of a human infectious virus spread in the human interconnected physical complex network. The findings extend previous empirical studies and could provide more strict constraints for current mathematical and physical modeling studies, such as the SIR model and its variants based on the theory of complex networks. [ABSTRACT FROM AUTHOR]

Precipitation modeling holds significant importance in various fields such as agriculture, animal husbandry, weather derivatives, hydrology, and risk and disaster preparedness. Stochastic precipitation generators (SPGs) represent a class of statistical models designed to generate synthetic data capable of simulating dry and wet precipitation stretches for a long duration. The construction of Hidden Markov Models (HMMs), which treat latent meteorological circumstances as hidden states, is an efficient technique for simulating precipitation. Considering that there are many choices of emission distributions used to generate positive precipitation, the characteristics of different distributions for simulating positive precipitation have not been fully explored. The paper includes a simulation study that demonstrates how the Pareto distribution, when used as the distribution for generating positive precipitation, addresses the limitations of the exponential and gamma distributions in predicting heavy precipitation events. Additionally, the Pareto distribution offers flexibility through adjustable parameters, making it a promising option for precipitation modeling. We can estimate parameters in HMMs using forward–backward algorithms, Variational Bayes Expectation-Maximization (VBEM), and Stochastic Variational Bayes (SVB). In the Xilingol League, located in the central part of the Inner Mongolia Autonomous Region, China, our study involved data analysis to identify crucial locations demonstrating a robust correlation and notable partial correlation between the Normalized Difference Vegetation Index (NDVI) and annual precipitation. We performed fitting of monthly dry days ratios and monthly precipitation using seasonal precipitation and year-round precipitation data at these crucial locations. Subsequently, we conducted precipitation predictions for the daily, monthly, and annual time frames using the new test dataset observations. The study concludes that the SPG fits the monthly dry-day ratio better for annual daily precipitation data than for seasonal daily precipitation data. The fitting error for the monthly dry day ratio corresponding to annual daily precipitation data is 0.053 (exponential distribution) and 0.066 (Pareto distribution), while for seasonal daily precipitation data, the fitting error is 0.14 (exponential distribution) and 0.15 (Pareto distribution). The exponential distribution exhibits the poorest performance as a model for predicting future precipitation, with average errors of 2.49 (daily precipitation), 40.62 (monthly precipitation), and 130.40 (annual precipitation). On the other hand, the Pareto distribution demonstrates the best overall predictive performance, with average errors of 0.69 (daily precipitation), 34.69 (monthly precipitation), and 66.42 (annual precipitation). The results of this paper can provide decision support for future grazing strategies in the Xilingol League. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Marshall–Olkin Pareto type-I (MOPTI) distribution. Statistical attributes of the MOPTI distribution including the quantile function, mean residual life, and a new theorem for strength-stress measure are introduced. Five methods of estimation for the MOPTI parameters based on complete samples are presented. Furthermore, we explore the estimation of the MOPTI parameters under type-I and type-II censoring. Two Monte Carlo simulation studies are conducted to evaluate the performance of the estimation methods under complete and censored samples. A real-life data set is used to validate the proposed methods. [ABSTRACT FROM AUTHOR]

Accurate parameter estimation is essential for modeling the statistical characteristics of ocean clutter. Common parameter estimation methods in generalized Pareto distribution models have limitations, such as restricted parameter ranges, lack of closed-form expressions, and low estimation accuracy. In this study, the particle swarm optimization (PSO) algorithm is used to solve the non-closed-form parameter estimation equations of the generalized Pareto distribution. The goodness-of-fit experiments show that the PSO algorithm effectively solves the non-closed parameter estimation problem and enhances the robustness of fitting the generalized Pareto distribution to heavy-tailed oceanic clutter data. In addition, a new parameter estimation method for the generalized Pareto distribution is proposed in this study. By using the difference between the statistical histogram of the data and the probability density function/cumulative distribution function of the generalized Pareto distribution as the target, an adaptive function with weighted coefficients is constructed to estimate the distribution parameters. A hybrid PSO (HPSO) algorithm is used to search for the best position of the fitness function to achieve the best parameter estimation of the generalized Pareto distribution. Simulation analysis shows that the HPSO algorithm outperforms the PSO algorithm in solving the parameter optimization task of the generalized Pareto distribution. A comparison with other traditional parameter estimation methods for generalized Pareto distribution shows that the HPSOHPSO algorithm exhibits strong parameter estimation performance, is efficient and stable, and is not limited by the parameter range. [ABSTRACT FROM AUTHOR]

The discrete Pareto (DP) distribution studied in this paper is a probability model with a power-law tail, which provides a convenient alternative to the well-known Zipf distribution. While basic characteristics of the DP model are available explicitly, this is not an exponential family and parameter estimation connected with this model is a challenging task. With this in mind we develop a computational approach to this problem, based on the expectation-maximization (EM) algorithm. In the process, we discover an interesting new probability distribution, which is a certain tilted version of the standard gamma model, and we provide a short account of its basic properties. The latter play a crucial role in our EM algorithm. Our computational approach to DP parameter estimation is illustrated by simulations, while a real data example from finance illustrates potential applications of the DP stochastic model. [ABSTRACT FROM AUTHOR]

The Moore-Bilikam distribution is convenient for survival analysis. The estimation of its parameters and its reliability function is performed by maximum likelihood, expectation-maximization, stochastic expectation-maximization, and the Bayesian method. The data are progressively censored of type II (samples are removed randomly from the experiment). Simulation shows that the expectation-maximization estimator of the parameter and the Bayesian-shrinkage estimator of the reliability function are the most efficient (with the minimum mean square error) when they are based on the Weibull and the Pareto distributions, which are specific cases of the Moore-Bilikam distribution. Bayesian and maximum likelihood estimations using the Moore-Bilikam distribution under type-II progressive censoring allow for fitting empirical failure times of an insulating fluid between two electrodes and the resistance of single carbon fibers. The associated reliability functions are estimated by each method. [ABSTRACT FROM AUTHOR]

In this article, we study a threshold leverage stochastic volatility model with a generalized hyperbolic skew Student's t distribution (TLSV-GHST). The model can simultaneously capture the leverage effect, skewness, and heavy-tailedness of financial return data. A popular Bayesian method combining the Metropolis-Hastings (MH) algorithm and the Gibbs sampler is developed for the parameter estimation of the TLSV-GHST model. The deviance information criterion is used to assess the fitness of the proposed model. Additionally, we investigate the sensitivity of Bayesian estimates to the priors of parameters. Some simulation studies and examples are presented to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A jointly censored sample is a very useful sampling technique in conducting comparative life tests of the products, its efficiency appears in permitting the selection of two samples from two manufacturing lines at the same time and conducting a life-testing experiment. This article presents estimation information of the joint generalized Pareto distributions parameters using Type-II progressive censoring scheme, which is carried out with binomial removal. The generalized Pareto distribution has many applications in different fields. We outline the problem of parameter estimation using the frequentest maximum likelihood and the Bayesian estimation methods. Furthermore, different interval estimation methods for estimating the four parameters were used: the asymptotic property of the maximum likelihood estimator, the credible confidence intervals, and the Bootstrap confidence intervals. The detailed numerical simulations have been considered to compare the performance of the proposed estimates. In addition, the applicability of the joint generalized Pareto censored model has been performed by applying a real data example. [ABSTRACT FROM AUTHOR]

This article analyzes six probability distributions from the Generalized Pareto family, with three, four and five parameters, with the main purpose of identifying other distributions from this family with applicability in flood frequency analysis compared to the distribution already used in the literature from this family such as Generalized Pareto Type II and Wakeby. This analysis is part of a larger and more complex research carried out in the Faculty of Hydrotechnics regarding the elaboration of a norm for flood frequency analysis using the linear moments method. In Romania, the standard method of parameter estimation is the method of ordinary moments, thus the transition from this method to the method of linear moments is desired. All the necessary elements for the distribution use are presented, such as the probability density functions, the complementary cumulative distribution functions, the quantile functions, and the exact and approximate relations for estimating parameters, for both methods of parameter estimation. All these elements are necessary for a proper transition between the two methods, especially since the use of the method of ordinary moments is done by choosing the skewness of the observed data depending on the origin of the maximum flows. A flood frequency analysis case study, using annual maximum and annual exceedance series, was carried out for the Prigor River to numerically present the analyzed distributions. The performance of this distribution is evaluated using a linear moments diagram. [ABSTRACT FROM AUTHOR]

Methods of estimating the parameters of generalized linear models for the case of paying insurance premiums to clients are considered. The iterative-recursive weighted least squares method, the Adam optimization algorithm, and the Monte Carlo method for Markov chains were implemented. Insurance indicators and the target variable were randomly generated due to the problem of public access to insurance data. For the latter, the normal and exponential law of distribution and the Pareto distribution with the corresponding link functions were used. Based on the quality metrics of model learning, conclusions were made regarding their construction quality. [ABSTRACT FROM AUTHOR]

All the available information and uncertainties should be taken into account in a model to give correct answer to a stated problem and evaluate the performance of a structure. This study deals with the impact of parameter estimation uncertainty in extreme wind speeds on the assessed reliability index using frequentist approach. The peak-over-threshold approach with an automated threshold selection method is applied and bootstrapping is used to determine the 95% confidence interval of the estimated reliability index. Based on the results practical recommendations, i.e., a framework of this procedure are derived on how to handle peak-over-threshold in extreme wind speed models for the application of the proposed performance-based wind engineering design. [ABSTRACT FROM AUTHOR]

Estimation based on probability-weighted moments is a well-established method and an excellent alternative to the classic method of moments or the maximum likelihood method, especially for small sample sizes. In this research, we developed a new class of estimators for the parameters of the Pareto type I distribution. A generalization of the probability-weighted moments approach is the foundation for this new class of estimators. It has the advantage of being valid in the entire parameter space of the Pareto distribution. We established the asymptotic normality of the new estimators and applied them to simulated and real datasets in order to illustrate their finite sample behavior. The results of comparisons with the most used estimation methods were also analyzed. [ABSTRACT FROM AUTHOR]

Exceedance analysis aims to model extreme data that are in the tail of the distribution, with these values being greater than a threshold. In this type of modelling, the tail can usually have different types of shapes. For exceedance, the extreme value theory shows that generalized Pareto distribution (GPD) is the limit distribution of sufficiently large thresholds. The studies on extreme analysis primarily focus on estimating the probability of the occurrence of extreme events, which are the values that are in the highest quantiles of the distribution. Based on this, this work proposed a quantile regression model for the GPD distribution, where the high quantiles of the distribution are written as a function of covariates. The proposed model also allows the regression parameters to vary over time. The parameter estimation of these new models is conducted under the Bayesian paradigm. Simulations are conducted to analyze the performance of our proposed models. Results of temperature applications in the USA and Brazil (environmental data) and consumer price index in Brazil (financial data) have efficiently explained how covariates influence the quantile behaviour over time. [ABSTRACT FROM AUTHOR]

In this paper, we consider several families of closed-form estimators of the two parameters of the Generalized Pareto Distribution (GPD). These estimators are easy to compute and have high efficiency when compared to previously proposed methods. We also consider some estimators which are not of closed-form. All methods are based on certain order statistics. The proposed procedures are best for extreme values of the shape parameters and sample sizes of 100 or larger. Monte Carlo simulations are conducted to investigate the performance of the proposed parameter estimation procedures. Our findings suggest that the proposed estimation methods are competitive compared to the existing methods. We provide a real data application to illustrate the utilization of the proposed methods in estimating the GPD parameters. [ABSTRACT FROM AUTHOR]

Various techniques of scale parameter estimation have been proposed in the case of alpha stable distributions. In the paper, the authors present an estimation technique that involves the k-th record theory. Although this theory is over 40 years old, its implementation in the classical extreme value theory – being the other cornerstone of the presented approach – is quite new, and tempting. Several theoretical properties of the introduced scale parameter estimators are presented. With the use of Monte Carlo methods, a comparative analysis is performed between the approach based on k-th records and approaches based on Hill's and Pickands' estimators. Additionally, the paper uses a real-life data set to illustrate how to effectively apply the k-th record estimator of the scale parameter. The research indicates several advantages of the k-th record approach over its other counterparts, especially when dealing with incomplete information about the underlying sample. [ABSTRACT FROM AUTHOR]

Considering an estimation problem of a shape parameter in a Pareto distribution under a restriction of a scale parameter, an estimator dominating all the previous estimators is proposed. Especially, under the quadratic loss function, the proposed estimator is shown to be second-order admissible in the class of all modified maximum likelihood estimators. [ABSTRACT FROM AUTHOR]

In this paper, the discrete exponentiated Pareto distribution with Bayesian parameter estimation is introduced. Some statistical properties of this distribution are discussed. We employed Bayesian approach for model parameter estimation. Simulation study is conducted for illustration of its estimation performance where it is evaluated according to the root of mean square error. The results indicated that the Bayesian approach reasonably provides parameter estimates that is nearly close to the predetermined parameter values when the sample size is increased. In addition, the proposed distribution with Bayesian estimates is significantly fitted to real dataset. [ABSTRACT FROM AUTHOR]

In this paper, we consider estimation of unknown parameters of the tapered Pareto distribution, which belongs to the class of semiheavy distributions, by a sample with excluded ℓn largest and kn smallest observations. We establish necessary and sufficient conditions in terms of proportions kn/n and ℓn/n for weak consistency and joint asymptotic normality of parameterizedmoment-type estimators for the shape and form parameters. Additionally, we extend the result on weak consistency of generalized Hill statistics (introduced in [V. Paulauskas and M. Vaičiulis, On the improvement of Hill and some others estimators, Lith. Math. J., 53(3):336–355, 2013]) to the case where the extreme value index is not positive. We demonstrate the performance of the proposed estimators on both simulated data from the tapered Pareto distribution and real data from finance. [ABSTRACT FROM AUTHOR]

In this article, a spatial-temporally varying generalized Pareto intensity distribution (STV-GPID) model is presented to characterize large-scene sea clutter in high-resolution maritime surveillance radars, and a multiscan recursive Bayesian bipercentile (MSRB-BiP) estimation method is proposed to implement the outlier-robust estimation of parameters in the STV-GPID model. Considering that sea clutter characteristics are affected by sea states and the viewing geometry of a radar, the large scene is segmented into clutter map cells based on an empirical backscattering coefficient model of sea surface to predict radar cross section (RCS) of sea surface per unit physical area. Sea clutter intensities on each clutter map cell are modeled by a GPID. In the parameter estimation, the data of previous scans are transformed into the prior information on the parameters to reduce the storage burden of radar systems. The MSRB-BiP estimator updates the parameters of the STV-GPID model recursively by a mixed sample set with the returns of the present scan and simulated data using the prior information. The mixture ratio adjusts the forgetting rate of data to adapt to temporally varying characteristics of sea clutter. At least, it brings three merits: low storage requirement, outlier robustness, and mitigation of spatial small sample size of a single scan. The convergence and robustness of the estimation method are verified by simulated data. The experimental results on two measured radar datasets verify the effectiveness of the MSRB-BiP estimators, and the errors at the steady state are reduced at least 17.7% and 66.7%, respectively. [ABSTRACT FROM AUTHOR]

The generalized Pareto (GP) distribution model is often used to describe the amplitude statistical feature of sea clutter. Generally, the parameters of GP distribution are estimated by moments estimators. However, when the sea state is high, the appearance of sea spikes will increase the echo of the anomalous scattering units, which leads to a decrease in the parameter estimation accuracy and target detection performance. To improve the parameter estimation accuracy, this paper proposes a novel parameter estimation method based on variable intervals. Considering the local properties of sea clutter, we take a variable interval of the entire sea clutter series for parameter estimation, where the interval position is selected according to the sea state conditions. For contrast, the bipercentile parameter estimation and truncate moment estimation are also introduced. Finally, the experiment based on the real measured X-band sea clutter datasets indicates that the proposed method outperforms the state-of-the-art moments estimators. [ABSTRACT FROM AUTHOR]

Estimation based on probability-weighted moments is a well-established method and an excellent alternative to the classic method of moments or the maximum likelihood method, especially for small sample sizes. In this research, we developed a new class of estimators for the parameters of the Pareto type I distribution. A generalization of the probability-weighted moments approach is the foundation for this new class of estimators. It has the advantage of being valid in the entire parameter space of the Pareto distribution. We established the asymptotic normality of the new estimators and applied them to simulated and real datasets in order to illustrate their finite sample behavior. The results of comparisons with the most used estimation methods were also analyzed.

Inverted version of exponentiated distributions are found useful as lifetime models. In this paper, we develop maximum likelihood estimators and Bayes estimators of unknown parameters of the inverted exponentiated Pareto distribution based on upper record values. We derive Bayes estimators of the parameters using importance sampling under symmetric and asymmetric loss functions. Credible interval estimation for the parameters is also done. A prediction for future upper record values is proposed based upon non-Bayesian as well as Bayesian approach. Bayes predictive interval estimation is carried out for future upper record value. We assess the validity of the proposed method by using real data and compare the proposed estimators based on mean squared error through a Monte Carlo simulation. The results may be of interest especially when only record values are stored. [ABSTRACT FROM AUTHOR]

The class of inhomogeneous phase‐type distributions (IPH) was recently introduced in Albrecher & Bladt (2019) as an extension of the classical phase‐type (PH) distributions. Like PH distributions, the class of IPH is dense in the class of distributions on the positive halfline, but leads to more parsimonious models in the presence of heavy tails. In this paper we propose a fitting procedure for this class to given data. We furthermore consider an analogous extension of Kulkarni's multivariate PH class (Kulkarni, 1989) to the inhomogeneous framework and study parameter estimation for the resulting new and flexible class of multivariate distributions. As a by‐product, we amend a previously suggested fitting procedure for the homogeneous multivariate PH case and provide appropriate adaptations for censored data. The performance of the algorithms is illustrated in several numerical examples, both for simulated and real‐life insurance data. [ABSTRACT FROM AUTHOR]

It is well known that inference for the generalized Pareto distribution (GPD) is a difficult problem since the GPD violates the classical regularity conditions in the maximum likelihood method. For parameter estimation, most existing methods perform satisfactorily only in the limited range of parameters. Furthermore, the interval estimation and hypothesis tests have not been studied well in the literature. In this article, we develop a novel framework for inference for the GPD, which works successfully for all values of shape parameter k. Specifically, we propose a new method of parameter estimation and derive some asymptotic properties. Based on the asymptotic properties, we then develop new confidence intervals and hypothesis tests for the GPD. The numerical results are provided to show that the proposed inferential procedures perform well for all choices of k. [ABSTRACT FROM AUTHOR]

The article deals with the problem of estimating the parameters of a tapered Pareto distribution. Using the moment method, we obtain new estimates depending on an additional parameter. We prove that the joint asymptotic distribution of these estimates is Gaussian. A procedure is proposed that permits one to choose the additional parameter in an optimal way. The new estimates are compared with the corresponding maximum likelihood estimates. By way of example, an application of the new estimates to the COVID-19 incidence data is given. A new algorithm for a random variable generator with a tapered Pareto distribution is proposed. [ABSTRACT FROM AUTHOR]

This article deals with different methods of point estimation for the unknown parameters of Marshall-Olkin Pareto distribution (MOP). This is a new lifetime that generalizes Pareto distribution, which was introduced by Marshall-Olkin (1997). Some classical point estimation methods are considered and their asymptotic properties are discussed along with studying Bayesian estimation method. The main purpose of this work is to determine which estimation method is more efficient under MOP distribution based on minimum average relative mean square error (MSE). Real data analyses are performed and it has been shown that MOP distribution is a better fit than the original Pareto distribution. In this paper, we compare the performances of these procedures through extensive numerical simulations. [ABSTRACT FROM AUTHOR]

In this study, we model a heterogeneous population assuming the three-component mixture of the Pareto distributions assuming type I censored data. In particular, we study some statistical properties (such as various entropies, different inequality indices, and order statistics) of the three-component mixture distribution. The ML estimation and the Bayesian estimation of the mixture parameters have been performed in this study. For the ML estimation, we used the Newton Raphson method. To derive the posterior distributions, different noninformative priors are assumed to derive the Bayes estimators. Furthermore, we also discussed the Bayesian predictive intervals. We presented a detailed simulation study to compare the ML estimates and Bayes estimates. Moreover, we evaluated the performance of different estimates assuming various sample sizes, mixing weights and test termination times (a fixed point of time after which all other tests are dismissed). The real-life data application is also a part of this study. [ABSTRACT FROM AUTHOR]

In this paper, we use the combined-unified hybrid censoring samples to obtain the maximum likelihood estimates of the unknown parameters, survival, and hazard functions of Pareto distribution. Next, we discuss some efficiency criteria of the maximum likelihood estimators, including; the unbiasedness, consistency, and sufficiency. Additionally, we use MCMC to obtain the Bayesian estimates of the unknown parameters. In addition, we calculate the intervals estimation of the unknown parameters. Finally, we analyze a set of real data in view of the theoretical findings of the paper. [ABSTRACT FROM AUTHOR]

The current article evaluates least-squares-based approaches for estimating parameters of the two-parameter Pareto distribution. The algebraic expressions for least squares (LS), relative least squares (RLS) and weighted least squares (WLS) estimators are derived by generating empirical cumulative distribution function (CDF) using mean rank, median rank and symmetrical CDF methods. The performance of the estimation approaches is evaluated through Monte Carlo simulations for different combinations of parameter values and sample sizes. The performance of the regression-based methods is then compared with one another and with the traditional maximum likelihood (ML) estimation method. Our simulation results unveil that among the regression-based methods, RLS has an improved or better performance compared to the other two regression-based approaches for samples of all sizes. Moreover, RLS performs better than the ML method for small samples. Among the rank methods used for generating empirical CDF, it is observed that the mean rank method outperformed other two rank methods. The simulation results are further corroborated by the application of all the methods on two real-life datasets representing damages caused by natural catastrophes. [ABSTRACT FROM AUTHOR]

Cost effective sampling is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the scale and shape parameters θ and α from p (θ , α) . Several traditional estimators and ad hoc estimators will be studied under MERSS. The estimators under MERSS are compared to the corresponding ones under SRS. The simulation results show that the estimators under MERSS are significantly more efficient than the ones under SRS. A real data set is used for illustration. [ABSTRACT FROM AUTHOR]

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The Pareto model, first used in socioeconomic problems, has successfully been applied in many other areas such as astronomy, biology, bibliometrics, demography, insurance, or risk management. Although there are several variants of this distribution, the current study focuses on the classic Pareto distribution, also known as the Pareto type I distribution. We propose a new class of estimators for the Pareto shape parameter, obtained through a modification of the probability weighted moment method, called the log generalized probability weighted moments method. In addition to the asymptotic distribution, Monte Carlo simulations were performed to analyze the finite sample behavior of the proposed new estimators. A comparison with the most used estimators, such as the moment, the maximum likelihood, the least squares, and the probability weighted moments estimators was also performed. In addition, the estimators were used to construct asymptotic confidence intervals. To illustrate an application of the different estimation methods to a real data set from a clinical trial complete the article. Results indicate an overall good performance of the new proposed class. [ABSTRACT FROM AUTHOR]

The generalized Pareto distribution (GPD) is a family of continuous distributions used to model the tail of the distribution to values higher than a threshold u. Despite the advantages of the GPD representation, its shape and scale parameters do not correspond to the expected value, which complicates the interpretation of regression models specified using the GPD. This study proposes a linear regression model in which the response variable is a GPD, using a new parametrization that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive real line response variable, as is usual in the context of generalized linear models. Furthermore, we propose a model for extreme values, in which the GPD parameters (mean and precision) are defined on the basis of a dynamic linear regression model. The novelty of the study lies in the time variation of the mean and precision parameter of the resulting distribution. The parameter estimation of these new models is performed under the Bayesian paradigm. Simulations are conducted to analyze the performance of our proposed models. Finally, the models are applied to environmental datasets (temperature datasets), illustrating their capabilities in challenging cases in extreme value theory. [ABSTRACT FROM AUTHOR]

Multi-lane factor (MLF) is a probability reduction reflecting unfavorable traffic loads over multiple lanes acting simultaneously on the most adverse position of a bridge. It is one of the key components of traffic load models for bridges. The most recent research established a multi-coefficient MLF model that clearly illustrated the lane load disparity and the probability reduction of their simultaneous actions. However, it used the block maxima (BM) method for extreme value modeling, which requires a large amount of traffic data. This study aims to adopt the peaks-over-threshold (POT) method to obtain more information from short-term traffic data and model the extreme coincident lane load effects (LLEs) for multi-coefficient MLF calibration. First, the multi-coefficient MLF model was reviewed. Thereafter, the bivariate POT method for coincident LLEs modeling using generalized Pareto distribution was proposed and formulated. Critical issues such as bivariate threshold selection and parameter estimation were addressed. Numerical examples were demonstrated to verify and validate the approach. Finally, the proposed approach was applied for calibrating the MLF of an experimental site with four traffic lanes. The results indicated that the coincident LLEs modeling using the POT approach was accurate and more effective than using the BM method when applied to limited data. The calibrated MLFs from the experimental site effectively revealed the lane load disparity of traffic loads over multiple lanes, which is not involved in the traffic load models of current bridge design specifications. Furthermore, the influence of other problems such as weight restriction on coincident LLEs modeling and MLF calibration were discussed. The proposed technique provides a sound approach for multi-coefficient MLF calibration of bridge assessment with short-term site-specific traffic data. [ABSTRACT FROM AUTHOR]

Ranked set sampling (RSS) is an efficient method for estimating parameters when exact measurement of observation is difficult and/or expensive. In the current paper, several traditional and ad hoc estimators of the scale and shape parameters θ and α from the Pareto distribution p (θ , α) will be respectively studied in cases when one parameter is known and when both are unknown under simple random sampling, RSS and some of its modifications such as extreme RSS(ERSS) and median RSS(MRSS). It is found for estimating of θ from p (θ , α) in which α is known, the best linear unbiased estimator (BLUE) under ERSS is more efficient than the other estimators under the other sampling techniques. For estimating of α from p (θ , α) in which θ is known, the modified BLUE under MRSS is more efficient than the other estimators under the other sampling techniques. For estimating of θ and α from p (θ , α) in which both are unknown, the ad hoc estimators under ERSS are more efficient than the other estimators under the other sampling techniques. All efficiencies of these estimators are simulated under imperfect ranking. A real data set is used for illustration. [ABSTRACT FROM AUTHOR]

In this paper, the derivation of the likelihood function for parameter estimation based on double ranked set sampling (DRSS) designs used by Sabry et al. (2019) for the estimation of the parameters of the power generalized Weibull distribution is considered. The developed likelihood function is then used for the estimation of the exponential Pareto distribution parameters. The maximum likelihood estimators (MLEs) are then investigated and compared to the corresponding ones based on simple random sampling (SRS) and ranked set sampling (RSS) designs. A Monte Carlo simulation is conducted and the absolute relative biases, mean square errors, and efficiencies are compared for the different designs. The relative efficiency of the DRSS estimates with respect to other designs was found to be higher in case of the exponential Pareto distribution (EP). [ABSTRACT FROM AUTHOR]

In this work, a new distribution called the Chen Pareto distribution was derived using the Chen-G family of distributions. The mixture representation of the distribution was obtained. Furthermore, some statistical properties such as moments, moment generating functions, order statistics properties of the distribution were explored. The parameter estimation for the distribution was done using the maximum likelihood estimation method and the performance of estimators was assessed by conducting an extensive simulation study. The distribution was applied to a real data set in which it performed best when compared to some related distributions. [ABSTRACT FROM AUTHOR]

This paper studies the E-Bayesian estimations and their E-MSE of Pareto distribution parameter under different loss functions. In order to measure the estimated error, in the case of the two hyper parameters, the definition of E-MSE (expected mean square error) is introduced based on the definition of E-Bayesian estimation, and the formulas of E-Bayesian estimation and the formulas of E-MSE are given, respectively. Monte Carlo simulations are performed to analysis the performances of the proposed methods, results are compared on the basis of E-MSE. Finally, combined with the golfers income problem are performed to calculated, the maximum likelihood estimations of the shape parameter and scale parameter are given, respectively, and that compared and analysed with the E-Bayesian estimations; moreover, the Bayesian credible intervals of the shape parameter and scale parameter are also given, respectively, by MCMC method. [ABSTRACT FROM AUTHOR]

This paper points out that equating the rate of exceedance over threshold to the probability of exceedance in the generalized Pareto distribution, as is often applied in practice, leads to erroneous model parameter estimation, under- or overestimation of hazard, and impairs the duality between the generalized Pareto (GPD) and the generalized extreme-value (GEV) distributions. The problem stems from the fundamental difference in the domain of definition: the rate of exceedance ∈ 0 , ∞ and the probability of exceedance ∈ 0 , 1 . The erroneous parameter estimation is a result of practice in model parameter estimation that uses the concept of 'return period' (the inverse of exceedance probability) for both the GEV and the GPD. By using the concept of 'average recurrence interval' (the inverse of exceedance rate) of extremes in stochastic processes, we illustrate that the erroneous hazard estimation of the GPD is resolved. The use of average recurrence interval along with the duality allows the use of either the GEV or GPD for extreme hazard analysis, regardless of whether the data are collected via block maxima or peaks over a threshold. Some recommendations with regard to the practice of distribution parameter estimation are given. We demonstrate the duality of the two distributions and the impact of using average recurrence interval instead of return period by analysis of wind gust data collected by an automatic weather station at Woomera, South Australia, Australia. [ABSTRACT FROM AUTHOR]

Nowadays, the use of probabilistic modeling for the design of engineering structures is becoming more and more widespread due to the advances in computer technology. In order to have a comprehensive picture about a meteorological phenomenon, e.g., wind actions on structures, uncertainties must be taken into account. From structural engineering and practical points of view, the effect of the length of short time series available for the analysis on the final results can be interesting to define a minimum observation-length. In this way, the real condition at the site can be utilized to assess wind loading effects on the structure. This paper deals with the effect of uncertainties associated with the parameter estimation and threshold selection. A four-year record of wind speed data of Sződliget is analyzed, and these results are compared with the results of neighboring sites, Penc, and Budapest. The peak over threshold (POT) method with maximum likelihood estimation are selected to obtain the basic wind velocity. The suitable threshold is chosen using an automatic threshold selection approach. According to our results, the applied automated threshold selection method provide reliable results, and it is simple and computationally inexpensive. This method may reduce associate errors of threshold selection in the future. It was found that at least approximately 100 realizations should exceed the specified threshold to earn reliable results. It means that 1-1.5-year and 4-year records of wind speeds are necessary for statistical inference in case of weakly dependent observations and for statistically independent events, respectively. [ABSTRACT FROM AUTHOR]

Eutrophication models are of great importance and are valuable tools for the development of policy and legislation. However, the parameter uncertainty and substantial computational cost lead to difficulties in decision-making, especially for complex models with multiple indicators. A multicriteria uncertainty analysis and parameter estimation (MUAPE) method, which selected behavioral parameters combined with Pareto domination and simultaneously obtained acceptable values for modeling by the maximum likelihood concept and kernel density estimation, was shown. This method, which did not assign thresholds and weights, was applied to analyze the uncertainty of the Chaohu Lake eutrophication model and estimate parameters. The results of the behavioral parameters were compared using different criterion sets, the relative error (RE) and the root mean square error (RMSE), and the results showed little discrepancy in terms of the effects on parameter uncertainty represented by the marginal probability density. The uncertainties of the parameters related to algal kinetics (i.e., BMR, PM, and KESS) were smaller than those of nutrient- and temperature-related parameters (i.e., KDN, Nitm, KTB, and KTHDR) for both sets of criteria. However, the reduction in the joint uncertainty of the two parameters was greater when RE was used than when RMSE was used. The acceptable values for the key parameters of the Chaohu Lake eutrophication model were also obtained by the RE criterion. The results strongly agreed with the observed values, and parameters could be applied for model prediction. This result indicated that the combination method was not only practical for reducing parameter uncertainty but also useful for determining parameter values. This method provides a basis for multicriteria uncertainty analysis and parameter estimation in eutrophication modeling. [ABSTRACT FROM AUTHOR]

This article deals with the classical and Bayesian estimation procedures of the parameters of inverse Pareto distribution (IPD) using randomly censored data. The Maximum likelihood and Bayes estimators of the parameters are derived. The asymptotic confidence intervals of the parameters based on observed Fisher information matrix are constructed. Bayes estimators of the parameters are derived under LINEX loss function using gamma informative priors. For Bayesian computations, Markov Chain Monte Carlo (MCMC) method is used. Also, highest posterior density credible intervals of the parameters based on MCMC techniques are constructed. A simulation study compares the performance of the various estimators. Finally, two real datasets are considered for illustrative purposes. [ABSTRACT FROM AUTHOR]

Confidence evaluation algorithms for the analysis of binary phase-shift keying (BPSK) signals are discussed. The phase of the product of the observed signal and constructed reference signal is extracted, and its spectrum is utilized to verify confidence. Two extreme statistics of the phase spectrum are developed for the confidence evaluation, which depend on goodness-of-fit tests based on the generalized extreme value distribution and generalized Pareto distribution respectively. Via extensive simulations, we verify the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In extreme value analysis, sensitivity of inference to the definition of extreme event is a paramount issue. Under the peaks-over-threshold approach, this translates directly into the need of fitting a Generalized Pareto distribution to observations above a suitable level that balances bias versus variance of estimates. Selection methodologies established in the literature face recurrent challenges such as an inherent subjectivity or high computational intensity. We suggest a truly automated method for threshold detection, aiming at time efficiency and elimination of subjective judgment. Based on the well-established theory of L-moments, this versatile data-driven technique can handle batch processing of large collections of extremes data, while also presenting good performance on small samples. The technique's performance is evaluated in a large simulation study and illustrated with significant wave height data sets from the literature. We find that it compares favorably to other state-of-the-art methods regarding the choice of threshold, associated parameter estimation and the ultimate goal of computationally efficient return level estimation. [ABSTRACT FROM AUTHOR]

The upper tail of a claim size distribution of a property line of business is frequently modelled by Pareto distribution. However, the upper tail does not need to be Pareto distributed, extraordinary shapes are possible. Here, the opportunities for the modelling of loss distributions are extended. The basic idea is the adjustment of a base distribution for their tails. The (generalised) Pareto distribution is used as base distribution for different reasons. The upper tail is in the focus and can be modelled well for special cases by a discrete mixture of the base distribution with a combination of the base distribution with an adapting distribution via the product of their survival functions. A kind of smoothed step is realised in this way in the original line function between logarithmic loss and logarithmic exceedance probability. The lower tail can also be adjusted. The new approaches offer the opportunity for stochastic interpretation and are applied to observed losses. For parameter estimation, a modification of the minimum Anderson Darling distance method is used. A new test is suggested to exclude that the observed upper tail is better modelled by a simple Pareto distribution. Q-Q plots are applied, and secondary results are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, kumaraswamy reciprocal family of distributions is introduced as a new continues model with some of approximation to other probabilistic models as reciprocal, beta, uniform, power function, exponential, negative exponential, weibull, rayleigh and pareto distribution. Some fundamental distributional properties, force of mortality, mills ratio, bowley skewness, moors kurtosis, reversed hazard function, integrated hazard function, mean residual life, probability weighted moments, bonferroni and lorenz curves, laplace-stieltjes transform of this new distribution with the maximum likelihood method of the parameter estimation are studied. Finally, four real data sets originally presented are used to illustrate the proposed estimators. [ABSTRACT FROM AUTHOR]