一个新的具有单调降失效率的寿命分布

本文由Pareto分布和Logarithmic分布"混合"生成两参数具有单调降失效率的新型寿命分布,研究了该分布的矩、熵、失效率函数、平均剩余寿命和参数的极大似然估计,应用EM算法求参数的极大似然估计,进行了数值模拟.     

Point estimates of parameters for Neuman distribution of order k

We construct point estimates of the parameters of a Neuman distribution of order k as a representative of the class of generalized Poisson distributions. The main properties of this distribution (a recurrence formula, cumulants and moments, derivatives with respect to parameters) are given in a system with infinitely many parameters, and the relationships are demonstrated with the previously obtained expressions in a two-parameter system. Among the point estimation methods we consider the moment method and the substitution method, which both lead to simple systems of equations; the solvability conditions for these systems are investigated. The efficiency of the estimators relative to the Cramer-Rao lower bound is examined and some conclusions are drawn regarding their applicability. The equations of the maximum likelihood estimation method are written out for infinitely many parameters and for the two-parameter case. __________ Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 93–109, 2008.     

Pareto严格稳定分布在保险理赔中的应用

本文研究了Pareto严格稳定分布在保险中的应用.利用极大似然估计的方法得到了Pareto严格稳定分布,正态分布和Pareto分布的参数估计.根据信息准则,表明Pareto严格稳定分布能够较好地拟合保险数据.     

Bayesian estimation ofg-and-k distributions using MCMC

Summary  In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of theg-and-k distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach. The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.     

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.     

帕累托索赔分布中风险参数的经验贝叶斯估计

本文建立了贝叶斯模型,讨论了帕累托索赔额分布中参数的估计问题,得到了风险参数的极大似然估计、贝叶斯估计和信度估计,并证明了这些估计的强相合性.在均方误差的意义下比较了这些估计的好坏,并通过数值模拟对均方误差进行了验证,结果表明,贝叶斯估计比其他估计具有较小的均方误差.最后,给出了结构参数的估计并证明了经验贝叶斯估计和经验贝叶斯信度估计的渐近最优性.     

Parameter estimation of inverse Lindley distribution for Type-I censored data

In life testing experiments, Type-I censoring scheme has been widely used due to its simplicity and poise with considerable gain in the completion time of an experiment. This article deals with the parameter estimation of inverse Lindley distribution when the data is Type-I censored. Estimates have been obtained under both the classical and Bayesian paradigm. In the classical scenario, estimates based on maximum likelihood and maximum product of spacings coupled with their 95% asymptotic confidence interval have been obtained. Under the Bayesian set up, the point estimate is obtained by considering squared error loss function using Markov Chain Monte Carlo technique and highest posterior density intervals based on these samples are reckoned. The performance of above mentioned techniques are evaluated on the basis of their simulated risks. Further, a real data set is analysed for appraisal of aforementioned estimation techniques under the specified censoring scheme.     

Estimating the parameters ofkth-order poisson distributions

Point estimators are considered for the two-parameter family ofkth-order Poisson distributions. A formula is derived for the lower bound on the estimate covariance matrix with a series-form information matrix, and the covariance matrix is calculated for characteristic parameter values. The relative efficiency of various estimation methods is analyzed (maximum likelihood method, method of moments, substitution method). Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 84–93, 1999.     

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.     

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).      

Estimation of Parameters of the Generalized Pareto\Distribution by the Least Squares

Traditional estimations of parameters of the generalized Pareto distribution (GPD) are generally constrained by the shape parameter of GPD. Such as: the method-of-moments (MOM), the probability-weighted moments (PWM), L-moments (LM), the maximum likelihood estimation (MLE) and so on. In this paper we use the fact that GPD can be transformed into the exponential distribution and use the results of parameters estimation for the exponential distribution, than we propose parameters estimators of the two-parameter or three-parameter GPD by the least squares method. Some asymptotic results are provided and the proposed method not constrained by the shape parameter of GPD. A simulation study is carried out to evaluate the performance of the proposed method and to compare them with other methods suggested in this paper. The simulation results indicate that the proposed method performs better than others in some common situation.     

Local polynomial maximum likelihood estimation for Pareto-type distributions

We discuss the estimation of the tail index of a heavy-tailed distribution when covariate information is available. The approach followed here is based on the technique of local polynomial maximum likelihood estimation. The generalized Pareto distribution is fitted locally to exceedances over a high specified threshold. The method provides nonparametric estimates of the parameter functions and their derivatives up to the degree of the chosen polynomial. Consistency and asymptotic normality of the proposed estimators will be proven under suitable regularity conditions. This approach is motivated by the fact that in some applications the threshold should be allowed to change with the covariates due to significant effects on scale and location of the conditional distributions. Using the asymptotic results we are able to derive an expression for the asymptotic mean squared error, which can be used to guide the selection of the bandwidth and the threshold. The applicability of the method will be demonstrated with a few practical examples.     

Strong Consistency of the Maximum Product of Spacings Estimates with Applications in Nonparametrics and in Estimation of Unimodal Densities

Distributions with unimodal densities are among the most commonly used in practice. However, for many unimodal distribution families the likelihood functions may be unbounded, thereby leading to inconsistent estimates. The maximum product of spacings (MPS) method, introduced by Cheng and Amin and independently by Ranneby, has been known to give consistent and asymptotically normal estimators in many parametric situations where the maximum likelihood method fails. In this paper, strong consistency theorems for the MPS method are obtained under general conditions which are comparable to the conditions of Bahadur and Wang for the maximum likelihood method. The consistency theorems obtained here apply to both parametric models and some nonparametric models. In particular, in any unimodal distribution family the asymptotic MPS estimator of the underlying unimodal density is shown to be universally L¹ consistent without any further conditions (in parametric or nonparametric settings).     

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.     

On maximum likelihood estimation of a Pareto mixture

In this paper we deal with maximum likelihood estimation (MLE) of the parameters of a Pareto mixture. Standard MLE procedures are difficult to apply in this setup, because the distributions of the observations do not have common support. We study the properties of the estimators under different hypotheses; in particular, we show that, when all the parameters are unknown, the estimators can be found maximizing the profile likelihood function. Then we turn to the computational aspects of the problem, and develop three alternative procedures: an EM-type algorithm, a Simulated Annealing and an algorithm based on Cross-Entropy minimization. The work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto distributions where all the parameters have to be estimated.     

Bayesian inference for Pr (Y

《Applied Mathematical Modelling》2014,38(5-6):1698-1709

Logspline Density Estimation for Censored Data

Abstract

Logspline density estimation is developed for data that may be right censored, left censored, or interval censored. A fully automatic method, which involves the maximum likelihood method and may involve stepwise knot deletion and either the Akaike information criterion (AIC) or Bayesian information criterion (BIC), is used to determine the estimate. In solving the maximum likelihood equations, the Newton–Raphson method is augmented by occasional searches in the direction of steepest ascent. Also, a user interface based on S is described for obtaining estimates of the density function, distribution function, and quantile function and for generating a random sample from the fitted distribution.     

Monte Carlo maximum likelihood circle fitting using circular density functions

Finding the “best-fitting” circle to describe a set of points in two dimensions is discussed in terms of maximum likelihood estimation. Several combinations of distributions are proposed to describe the stochastic nature of points in the plane, as the points are considered to have a common, typically unknown center, a random radius, and random angular orientation. A Monte Carlo search algorithm over part of the parameter space is suggested for finding the maximum likelihood parameter estimates. Examples are presented, and comparisons are drawn between circles fit by this proposed method, least squares, and other maximum likelihood methods found in the literature.     

Reliability Statistical Analysis of Two-parameter Exponential Distribution under Constant Stress Accelerated Life Test with Inverse Power Law Model

Based on the inverse power law model, the maximum likelihood estimation and interval estimation of two-parameter Exponential distribution are derived in detail under constant stress accelerated life test. Secondly, the accuracy of point estimation and interval estimation is investigated by a large number of Monte Carlo simulations. Finally, examples and simulation examples are given to illustrate the application of the proposed method.     

On the mean square error of maximum likelihood estimates of the distribution density of sufficient statistics of the multivariate normal distribution

The extensive use of maximum likelihood estimates underscores the importance of the problem of statistical estimation of their errors. These estimates are of utmost importance in cases where the family of normal distributions and the families related to the normal distributions are considered [1, 2, 4]. The mean square errors of the maximum likelihood estimates of the normal density were investigated in the author's paper [3]. The mean square errors of statistical estimates of some families of densities related to the normal distributions were considered in the papers [4–6]. In the present paper, we obtain an asymptotic expansion of the mean square error of the maximum likelihood estimates of the densities of the joint distribution of sufficient statistics of the family of multivariate normal distributions. The results obtained allow us to construct the mean square errors of the maximum likelihood estimates for the chi-square density and Wishart's density. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 4–11, Perm. 1990.