Moment Estimation for Multivariate Extreme Value Distribution in a Nested Logistic Model

This paper considers multivariate extreme value distribution in a nested logistic model. The dependence structure for this model is discussed. We find a useful transformation that transformed variables possess the mixed independence. Thus, the explicit algebraic formulae for a characteristic function and moments may be given. We use the method of moments to derive estimators of the dependence parameters and investigate the properties of these estimators in large samples via asymptotic theory and in finite samples via computer simulation. We also compare moment estimation with a maximum likelihood estimation in finite sample sizes. The results indicate that moment estimation is good for all practical purposes.     

单参数Pareto分布变点估计研究

研究单参数Pareto分布存在变点时的估计问题,分别利用极大似然估计法和贝叶斯方法对单参数Pareto分布的变点进行估计,并运用Matlab软件进行随机模拟,随机结果表明贝叶斯方法与极大似然估计相比,估计值更接近真值.     

Synoptic airflow and UK daily precipitation extremes

We develop a vector generalised linear model to describe the influence of the atmospheric circulation on extreme daily precipitation across the UK. The atmospheric circulation is represented by three covariates, namely synoptic scale airflow strength, direction and vorticity; the extremes are represented by the monthly maxima of daily precipitation, modelled by the generalised extreme value distribution (GEV). The model parameters for data from 689 rain gauges across the UK are estimated using a maximum likelihood estimator. Within the framework of vector generalised linear models, various plausible models exist to describe the influence of the individual covariates, possible nonlinearities in the covariates and seasonality. We selected the final model based on the Akaike information criterion (AIC), and evaluated the predictive power of individual covariates by means of quantile verification scores and leave-one-out cross validation. The final model conditions the location and scale parameter of the GEV on all three covariates; the shape parameter is modelled as a constant. The relationships between strength and vorticity on the one hand, and the GEV location and scale parameters on the other hand are modelled as natural cubic splines with two degrees of freedom. The influence of direction is parameterised as a sine with amplitude and phase. The final model has a common parameterisation for the whole year. Seasonality is partly captured by the covariates themselves, but mostly by an additional annual cycle that is parameterised as a phase-shifted sine and accounts for physical influences that we have not attempted to explicitly model, such as humidity.     

Confidence Intervals and Accuracy Estimation for Heavy-Tailed Generalized Pareto Distributions

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.     

具有随机右删失随机变量分位数的经验似然推断

本文研究了具有随机右删失随机变量分位数的置信域的构造.利用经验似然和截尾值估算相结合的方法,给出了分位数的对数经验似然比统计量,在较少的条件下证明了该统计量的极限分布为自由度为1的x~2分布.使得完全数据下的分位数的经验似然推断方法应用到非完全数据中.     

指数分布抽样基本定理及在三参数一般指数分布参数估计中的应用

讨论三参数一般指数分布的参数估计,首先讨论了三参数一般指数分布参数的最大似然估计的求解问题,当其中参数α=1时,应用指数分布抽样基本定理,得到了三参数一般指数分布其它参数的一致最小方差无偏估计;并且由此给出求解三参数一般指数分布参数最大似然估计的迭代方法,得到了三参数一般指数分布参数最大似然估计的近似值,给出了模拟结果以说明迭代方法的收敛性;并以相关文献的观察数据作为样本,得到了三参数一般指数分布的参数估计,从而说明了迭代方法的有效性.     

The Maximum Lq-Likelihood Method: An Application to Extreme Quantile Estimation in Finance

Estimating financial risk is a critical issue for banks and insurance companies. Recently, quantile estimation based on extreme value theory (EVT) has found a successful domain of application in such a context, outperforming other methods. Given a parametric model provided by EVT, a natural approach is maximum likelihood estimation. Although the resulting estimator is asymptotically efficient, often the number of observations available to estimate the parameters of the EVT models is too small to make the large sample property trustworthy. In this paper, we study a new estimator of the parameters, the maximum Lq-likelihood estimator (MLqE), introduced by Ferrari and Yang (Estimation of tail probability via the maximum Lq-likelihood method, Technical Report 659, School of Statistics, University of Minnesota, 2007 ). We show that the MLqE outperforms the standard MLE, when estimating tail probabilities and quantiles of the generalized extreme value (GEV) and the generalized Pareto (GP) distributions. First, we assess the relative efficiency between the MLqE and the MLE for various sample sizes, using Monte Carlo simulations. Second, we analyze the performance of the MLqE for extreme quantile estimation using real-world financial data. The MLqE is characterized by a distortion parameter q and extends the traditional log-likelihood maximization procedure. When q→1, the new estimator approaches the traditional maximum likelihood estimator (MLE), recovering its desirable asymptotic properties; when q ≠ 1 and the sample size is moderate or small, the MLqE successfully trades bias for variance, resulting in an overall gain in terms of accuracy (mean squared error).      

定时截尾下指数分布产品可靠性抽样检验方案

本文给出了制订定时截尾下指数分布产品可靠性抽样检验方案的统计方法.检验统计量是平均寿命倒数的极大似然估计.提出了一种选择截尾时间的方法.利用分布分位数的Cornish-Fisher展开近似地确定了样本量和接收常数.模拟结果表明,本文给出的方法是可行的.     

Fitting the three-parameter Weibull distribution with Cross Entropy

The Weibull distribution is widely used in applications such as reliability and lifetime studies. Although this distribution has three parameters, for simplicity, literature pertaining to Weibull parameter estimation relaxes one of its parameters in order to estimate the other two. When the three-parameter Weibull distribution is of interest, the estimation procedure is complicated. For example, the likelihood function for a three-parameter Weibull distribution is hard to maximize. In this paper, a Cross Entropy (CE) method is developed in the context of maximum likelihood estimation (MLE) of a three-parameter Weibull distribution. Performing a simulation study, a comparative analysis between the newly developed method and two existing methods is conducted. The results show the proposed method has better performance in terms of accuracy, precision and run time for different parameter settings and sample sizes.     

双边定时截尾样本下广义逆指数分布形状参数的估计

先给出了广义逆指数分布在双边定时截尾样本下形状参数的最大似然估计,并不能得到估计的显式表达式,但证明了参数在(0,+∞)上最大似然估计是唯一存在的.其次提出用EM算法求出形状参数的估计且该估计具有良好的收敛性,还给出了形状参数的EM估计的渐近方差和近似置信区间;最后通过数值模拟,对形状参数的最大似然估计和EM估计的效果进行了比较,说明了用EM算法求形状参数的估计是可行的,并且模拟效果相对比较好.     

Pareto分布环境因子的估计及其应用

给出了Pareto分布环境因子的定义,讨论了在定数截尾样本下Pareto分布环境因子的极大似然估计和修正极大似然估计,并尝试把环境因子用于可靠性评估中.最后运用Monte Carlo方法对极大似然估计,修正极大似然估计和可靠性指标的均方误差(MSE),进行了模拟比较,结果表明修正极大似然估计优于极大似然估计且考虑环境因子的可靠性评估结果较好.     

具有缺失数据的两个伽马分布总体参数的估计与假设检验

主要在数据缺失的情况下研究了伽马分布的参数估计与假设检验,位置参数已知的条件下,给出形状参数的极大似然估计,并证明了形状参数估计的强相合性与渐进正态性,并对两总体参数之差的置信区间和假设检验做出分析,最后做随机模拟验证了其合理性.     

Regular Score Tests of Independence in Multivariate Extreme Values

The score tests of independence in multivariate extreme values derived by Tawn (Tawn, J.A., “Bivariate extreme value theory: models and estimation,” Biometrika 75, 397–415, 1988) and Ledford and Tawn (Ledford, A.W. and Tawn, J.A., “Statistics for near independence in multivariate extreme values,” Biometrika 83, 169–187, 1996) have non-regular properties that arise due to violations of the usual regularity conditions of maximum likelihood. Two distinct types of regularity violation are encountered in each of their likelihood frameworks: independence within the underlying model corresponding to a boundary point of the parameter space and the score function having an infinite second moment. For applications, the second form of regularity violation has the more important consequences, as it results in score statistics with non-standard normalisation and poor rates of convergence. The corresponding tests are difficult to use in practical situations because their asymptotic properties are unrepresentative of their behaviour for the sample sizes typical of applications, and extensive simulations may be needed in order to evaluate adequately their null distribution. Overcoming this difficulty is the primary focus of this paper. We propose a modification to the likelihood based approaches used by Tawn (Tawn, J.A., “Bivariate extreme value theory: models and estimation,” Biometrika 75, 397–415, 1988) and Ledford and Tawn (Ledford, A.W. and Tawn, J.A., “Statistics for near independence in multivariate extreme values,” Biometrika 83, 169–187, 1996) that provides asymptotically normal score tests of independence with regular normalisation and rapid convergence. The resulting tests are straightforward to implement and are beneficial in practical situations with realistic amounts of data. AMS 2000 Subject Classification Primary—60G70 Secondary—62H15     

基于左截断右删失数据的单参数Pareto分布参数估计

基于EM算法及极大似然法研究了左截断右删失数据下单参数Pareto分布的参数估计,导出其迭代式,并应用随机模拟对参数估计式进行了模拟检验,结果表明迭代式能够快速收敛,EM估计值较为精确.     

Maximum Likelihood Estimation of Hidden Markov Multivariate Normal Distribution Parameters

Hidden Markov model is widely used in statistical modeling of time, space and state transition data. The definition of hidden Markov multivariate normal distribution is given. The principle of using cluster analysis to determine the hidden state of observed variables is introduced. The maximum likelihood estimator of the unknown parameters in the model is derived. The simulated observation data set is used to test the estimation effect and stability of the method. The characteristic is simple classical statistical inference such as cluster analysis and maximum likelihood estimation. The method solves the parameter estimation problem of complex statistical models.     

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??Hidden Markov model is widely used in statistical modeling of time, space and state transition data. The definition of hidden Markov multivariate normal distribution is given. The principle of using cluster analysis to determine the hidden state of observed variables is introduced. The maximum likelihood estimator of the unknown parameters in the model is derived. The simulated observation data set is used to test the estimation effect and stability of the method. The characteristic is simple classical statistical inference such as cluster analysis and maximum likelihood estimation. The method solves the parameter estimation problem of complex statistical models.     

Lomax分布极大似然估计的两点研究

本文研究了Lomax分布参数极大似然估计的存在性和估计量的收敛性问题.利用严格的分析法和中心极限定理,获得了Lomax分布极大似然估计的存在性和估计量的渐近正态分布的结果,进一步推广到了有缺失数据的两个Lomax总体中,参数的极大似然估计有强相合性和渐近正态性.     

On the maximum likelihood estimator for the Generalized Extreme-Value distribution

The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood estimator has never been established. In this paper, a formal proof is given using a general result on the maximum likelihood estimator for parametric families that are differentiable in quadratic mean but whose supports depend on the parameter. An interesting side result concerns the (lack of) differentiability in quadratic mean of the GEV family.     

Optimal estimation of a signal perturbed by a sub-fractional Brownian motion

We consider the problem of optimal estimation of the vector parameter θ of the drift term in a sub-fractional Brownian motion. We obtain the maximum likelihood estimator as well as Bayesian estimator when the prior distribution is Gaussian.     

有关顺序统计量的极大似然估计

讨论了一类参数空间受样本限制的极大似然估计问题.分析了随机变量分布的非零区域与似然函数定义域的对应关系,提出如果分布的非零区域受参数限制,则无论似然方程是否可解,参数的极大似然估计必然与样本顺序统计量X_((n))或X_((1))有关,并具体分析了似然估计一定等于、一定不等于和可能等于顺序统计量X_((n))(X_((1)))的三种情形,并给出了相应的判别条件.最后分析得出在第三种判别条件之下,似然估计是否取值于x_((n))(x_((1)))视具体的样本观测值决定.