Tests for the Validity of the Assumption that the Underlying Distribution of Life is Pareto

This article considers the problem of testing the validity of the assumption that the underlying distribution of life is Pareto. For complete and censored samples, the relationship between the Pareto and the exponential distributions could be of vital importance to test for the validity of this assumption. For grouped uncensored data the classical Pearson χ² test based on the multinomial model can be used. Attention is confined in this article to handle grouped data with withdrawals within intervals. Graphical as well as analytical procedures will be presented. Maximum likelihood estimators for the parameters of the Pareto distribution based on grouped data will be derived.     

Second-order refined peaks-over-threshold modelling for heavy-tailed distributions

Modelling excesses over a high threshold using the Pareto or generalized Pareto distribution (PD/GPD) is the most popular approach in extreme value statistics. This method typically requires high thresholds in order for the (G)PD to fit well and in such a case applies only to a small upper fraction of the data. The extension of the (G)PD proposed in this paper is able to describe the excess distribution for lower thresholds in case of heavy-tailed distributions. This yields a statistical model that can be fitted to a larger portion of the data. Moreover, estimates of tail parameters display stability for a larger range of thresholds. Our findings are supported by asymptotic results, simulations and a case study.     

Extended generalised Pareto models for tail estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.     

基于改进的Pareto/NBD模型预测博客用户在线行为

博客用户在线行为分为发文行为和流失行为.由于这两种行为分别与交易过程中客户的购买行为和流失行为具有相似性,选择借鉴客户基分析中的Pareto/NBD模型进行预测.考虑到用户间交互性对博客用户在线行为具有重要影响,通过比例风险模型向经典的Pareto/NBD模型中加入体现用户间交互性的协变量.Pareto/NBD模型经过改进,实现了对博客用户在线行为的预测.实证研究以用户博客空间中的总评论量和总浏览量作为协变量.数据分析结果显示,当使用总评论量作为影响流失行为的协变量时,改进模型的预测精度显著提高.进一步分析还发现,总评论量对博客用户“存活”时长的正向激励存在着阈值.     

Parameter estimation for generalized Pareto distribution by generalized probability weighted moment-equations

The generalized Pareto distribution (GPD) has been widely used to model exceedances over a threshold. This article generalizes the method of generalized probability weighted moments, and applies this method to estimate the parameters of GPD. The estimator is computationally easy. Some asymptotic results of this method are provided. Two simulations are carried out to investigate the behavior of this method and to compare them with other methods suggested in the literature. The simulation results show that the performance of the proposed method is better than some other methods. Finally, this method is applied to analyze a real-life data.     

Evaluation of a Frequentist Hierarchical Model to Estimate Prevalence When Sampling from a Large Geographic Area Using Pool Screening

We present a frequentist Bernoulli-Beta hierarchical model to relax the constant prevalence assumption underlying the traditional prevalence estimation approach based on pooled data. This assumption is called into question when sampling from a large geographic area. Pool screening is a method that combines individual items into pools. Each pool will either test positive (at least one of the items is positive) or negative (all items are negative). Pool screening is commonly applied to the study of tropical diseases where pools consist of vectors (e.g., black flies) that can transmit the disease. The goal is to estimate the proportion of infected vectors.

Intermediate estimators (model parameters) and estimators of ultimate interest (pertaining to prevalence) are evaluated by standard measures of merit, such as bias, variance, and mean squared error making extensive use of expansions. Using the hierarchical model an investigator can determine the probability of the prevalence being below a pre-specified threshold value, a value at which no reemergence of the disease is expected. An investigation into the least biased choice of the α parameter in the Beta (α, β) prevalence distribution leads to the choice of α = 1.     

中国商业银行内部欺诈风险度量研究

作为巴塞尔新资本协议规定的七种操作风险损失类型之一,内部欺诈问题是中国商业银行的一个重大风险来源。以部分国内商业银行内部欺诈数据为样本,针对内部欺诈具有的低频率高损失的特点,借助广义Pareto分布（GPD）和对数正态分布对内部欺诈建立了一个风险度量模型,然后通过对尾部分布何时服从GPD进行检验,得到了精确的门限值,最后利用所建立的分布模型对内部欺诈类操作风险在险风险值、经济资本和最大可能损失进行了估计,说明了中国商业银行防范内部欺诈风险的重要性。     

Regression models for time-varying extremes

A common approach to modelling extreme data are to consider the distribution of the exceedance value over a high threshold. This approach is based on the distribution of excess, which follows the generalized Pareto distribution (GPD) and has shown to be adequate for this type of situation. As with all data involving analysis in time, excesses above a threshold may also vary and suffer from the influence of covariates. Thus, the GPD distribution can be modelled by entering the presence of these factors. This paper presents a new model for extreme values, where GPD parameters are written on the basis of a dynamic regression model. The estimation of the model parameters is made under the Bayesian paradigm, with sampling points via MCMC. As with environmental data, behaviour data are related to other factors such as time and covariates such as latitude and distance from the sea. Simulation studies have shown the efficiency and identifiability of the model, and applying real rain data from the state of Piaui, Brazil, shows the advantage in predicting and interpreting the model against other similar models proposed in the literature.     

Non-identifiable parametric probability models and reparametrization

Identifiability is a primary assumption in virtually all classical statistical theory. However, such an assumption may be violated in a variety of statistical models. We consider parametric models where the assumption of identifiability is violated, but otherwise satisfy standard assumptions. We propose an analytic method for constructing new parameters under which the model will be at least locally identifiable. This method is based on solving a system of linear partial differential equations involving the Fisher information matrix. Some consequences and valid inference procedures under non-identifiability have been discussed. The method of reparametrization is illustrated with an example.     

A diagnostic for selecting the threshold in extreme value analysis

A new approach is suggested for choosing the threshold when fitting the Hill estimator of a tail exponent to extreme value data. Our method is based on an easily computed diagnostic, which in turn is founded directly on the Hill estimator itself, 'symmetrized' to remove the effect of the tail exponent but designed to emphasize biases in estimates of that exponent. The attractions of the method are its accuracy, its simplicity and the generality with which it applies. This generality implies that the technique has somewhat different goals from more conventional approaches, which are designed to accommodate the minor component of a postulated two-component Pareto mixture. Our approach does not rely on the second component being Pareto distributed. Nevertheless, in the conventional setting it performs competitively with recently proposed methods, and in more general cases it achieves optimal rates of convergence. A by-product of our development is a very simple and practicable exponential approximation to the distribution of the Hill estimator under departures from the Pareto distribution.     

对生产帕累托最优条件充分性的质疑与改进

现有的理论中,生产帕累托最优条件不充分,其最优状态不是唯一的,使得理论自身及随后的应用上产生了一些矛盾。确定唯一最优状态的充分条件应该是：双方都满意的交换利益分割和边际技术替代率相等。前者保证交换实现,后者保证交换产生的利益被完全穷尽。只要遵循新交换原则,交换必能实现,但不一定是帕累托最优。信息不充分时,如果信息不对称者对对方所判断的生产函数改变了其真实生产函数的技术性质,交换虽能实现,但生产并没有达到帕累托最优。     

Approximate studentization for Pareto distribution with application to censored data

The use of the Pareto distribution as a model for various socio-economic phenomena dates back to the late nineteenth century. Recently, it has also been recognized as a useful model for the analysis of lifetime data. In this paper, we apply the approximate studentization method to obtain inference for the scale parameter of the Pareto distribution, and also for the strong Pareto law. Moreover, we extend the method to construct prediction limits for thejth smallest future observation based on the firstk observed data.     

A semiparametric Bayesian approach to extreme value estimation

This paper is concerned with extreme value density estimation. The generalized Pareto distribution (GPD) beyond a given threshold is combined with a nonparametric estimation approach below the threshold. This semiparametric setup is shown to generalize a few existing approaches and enables density estimation over the complete sample space. Estimation is performed via the Bayesian paradigm, which helps identify model components. Estimation of all model parameters, including the threshold and higher quantiles, and prediction for future observations is provided. Simulation studies suggest a few useful guidelines to evaluate the relevance of the proposed procedures. They also provide empirical evidence about the improvement of the proposed methodology over existing approaches. Models are then applied to environmental data sets. The paper is concluded with a few directions for future work.     

A semi-parametric Bayesian extreme value model using a Dirichlet process mixture of gamma densities

In this paper, we propose a model with a Dirichlet process mixture of gamma densities in the bulk part below threshold and a generalized Pareto density in the tail for extreme value estimation. The proposed model is simple and flexible for posterior density estimation and posterior inference for high quantiles. The model works well even for small sample sizes and in the absence of prior information. We evaluate the performance of the proposed model through a simulation study. Finally, the proposed model is applied to a real environmental data.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

A Latent Process Model for Temporal Extremes

This paper presents a hierarchical approach to modelling extremes of a stationary time series. The procedure comprises two stages. In the first stage, exceedances over a high threshold are modelled through a generalized Pareto distribution, which is represented as a mixture of an exponential variable with a Gamma distributed rate parameter. In the second stage, a latent Gamma process is embedded inside the exponential distribution in order to induce temporal dependence among exceedances. Unlike other hierarchical extreme‐value models, this version has marginal distributions that belong to the generalized Pareto family, so that the classical extreme‐value paradigm is respected. In addition, analytical developments show that different choices of the underlying Gamma process can lead to different degrees of temporal dependence of extremes, including asymptotic independence. The model is tested through a simulation study in a Markov chain setting and used for the analysis of two datasets, one environmental and one financial. In both cases, a good flexibility in capturing different types of tail behaviour is obtained.     

地质灾害损失分布拟合与风险度量

 地质灾害的频繁发生已引起了社会各界的高度关注。本文以湖南省娄底市地质灾害损失数据为样本,借助广义Pareto分布和对数正态分布对地质灾害损失分布进行刻画,建立了一个分段的地质灾害损失分布模型,然后讨论了地质灾害损失的纯保费和最大可能损失的估计问题,得到了一些有意义的结果。     

A model for markers and latent health status

We extend the bivariate Wiener process considered by Whitmore and co-workers and model the joint process of a marker and health status. The health status process is assumed to be latent or unobservable. The time to reach the primary end point or failure (death, onset of disease, etc.) is the time when the latent health status process first crosses a failure threshold level. Inferences for the model are based on two kinds of data: censored survival data and marker measurements. Covariates, such as treatment variables, risk factors and base-line conditions, are related to the model parameters through generalized linear regression functions. The model offers a much richer potential for the study of treatment efficacy than do conventional models. Treatment effects can be assessed in terms of their influence on both the failure threshold and the health status process parameters. We derive an explicit formula for the prediction of residual failure times given the current marker level. Also we discuss model validation. This model does not require the proportional hazards assumption and hence can be widely used. To demonstrate the usefulness of the model, we apply the methods in analysing data from the protocol 116a of the AIDS Clinical Trials Group.     

Bayesian analysis for heteroscedastic normal-Pareto mixture model with application

ABSTRACT

In this article, we propose a new distribution by mixing normal and Pareto distributions, and the new distribution provides an unusual hazard function. We model the mean and the variance with covariates for heterogeneity. Estimation of the parameters is obtained by the Bayesian method using Markov Chain Monte Carlo (MCMC) algorithms. Proposal distribution in MCMC is proposed with a defined working variable related to the observations. Through the simulation, the method shows a dependable performance of the model. We demonstrate through establishing model under a real dataset that the proposed model and method can be more suitable than the previous report.     

Forecast of the expected non-epidemic morbidity of acute diseases using resampling methods

In epidemiological surveillance it is important that any unusual increase of reported cases be detected as rapidly as possible. Reliable forecasting based on a suitable time series model for an epidemiological indicator is necessary for estimating the expected non-epidemic indicator and to elaborate an alert threshold. Time series analyses of acute diseases often use Gaussian autoregressive integrated moving average models. However, these approaches can be adversely affected by departures from the true underlying distribution. The objective of this paper is to introduce a bootstrap procedure for obtaining prediction intervals in linear models in order to avoid the normality assumption. We present a Monte Carlo study comparing the finite sample properties of bootstrap prediction intervals with those of alternative methods. Finally, we illustrate the performance of the proposed method with a meningococcal disease incidence series.