A new method for selection of population distribution and parameter estimation

An optimization method for parameter estimation is presented with the Kolmogorov-Smirnov distance used as the objective. A step-by-step implementation procedure is given. The method is demonstrated by estimating the parameters for three-parameter Weibull distributions from three different samples (with different sample sizes). A comparison of the proposed method and the usual methods such as the least-squares method, the matching moments method and the maximum likelihood method shows that more reasonable estimates of the parameters are given by the proposed optimization method. Then, the proposed method is successfully extended to estimate the parameters for the sum of two three-parameter Weibull distributions. Based on these findings, a new procedure for selection of population distribution and parameter estimation is presented.     

On estimating Weibull modulus by moments and maximum likelihood methods

The effect of true Weibull modulus and sample size on Weibull modulus estimated by moments and maximum likelihood methods was investigated. Results indicated that the value of true Weibull modulus had no effect on estimated modulus for the maximum likelihood method, and a strong effect for the moments method, especially when sample size was less than 30. In addition, the distribution of Weibull modulus estimated by both methods was investigated using the modified Anderson–Darling statistics for goodness of fit. It was found that the distribution was not normal, lognormal, 3-parameter Weibull, or 3-parameter log-Weibull for the maximum likelihood method, as reported in previous studies. For the moments method however, the distribution of normalized Weibull moduli was found to be lognormal for sample sizes of 40 and above. The other three distributions showed a significant level of lack-of-fit at all sample sizes. An erratum to this article can be found at     

Maximum likelihood Estimation of Parameters in the Inverse Gaussian Distribution,With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.     

复合材料强度的统计分析 Ⅱ.参数估计

为了估计先前提出的修正的韦布尔分布函数的三个未知参数, 本文改改进了现有的作图估计方法, 还提出了一个联合估计方法, 经检验这些方法是适用的。      

Maximum Likelihood Estimation of the Parameters of the Gamma Distribution and Their Bias

The numerical technique of the maximum likelihood method to estimate the parameters of Gamma distribution is examined. A convenient table is obtained to facilitate the maximum likelihood estimation of the parameters and the estimates of the variance-covariance matrix. The bias of the estimates is investigated numerically. The empirical result indicates that the bias of both parameter estimates produced by the maximum likelihood method is positive.     

Generalized Marshall Olkin Inverse Lindley Distribution with Applications

In this article, a new generalization of the inverse Lindley distribution is introduced based on Marshall-Olkin family of distributions. We call the new distribution, the generalized Marshall-Olkin inverse Lindley distribution which offers more flexibility for modeling lifetime data. The new distribution includes the inverse Lindley and the Marshall-Olkin inverse Lindley as special distributions. Essential properties of the generalized Marshall-Olkin inverse Lindley distribution are discussed and investigated including, quantile function, ordinary moments, incomplete moments, moments of residual and stochastic ordering. Maximum likelihood method of estimation is considered under complete, Type-I censoring and Type-II censoring. Maximum likelihood estimators as well as approximate confidence intervals of the population parameters are discussed. A comprehensive simulation study is done to assess the performance of estimates based on their biases and mean square errors. The notability of the generalized Marshall-Olkin inverse Lindley model is clarified by means of two real data sets. The results showed the fact that the generalized Marshall-Olkin inverse Lindley model can produce better fits than power Lindley, extended Lindley, alpha power transmuted Lindley, alpha power extended exponential and Lindley distributions.     

Using partial probability weighted moments and partial maximum entropy to estimate quantiles from censored samples

The maximum entropy principle constrained by probability weighted moments is an useful technique for unbiasedly and efficiently estimating the quantile function of a random variable from a sample of complete observations. However, censored or incomplete data are often encountered in engineering reliability and lifetime distribution analysis. This paper presents a new distribution free method for the estimation of the quantile function of a non-negative random variable using a censored sample of data, which is based on the principle of partial maximum entropy (MaxEnt) in which partial probability weighted moments (PPWMs) are used as constraints. Numerical results and practical examples presented in the paper confirm the accuracy and efficiency of the proposed partial MaxEnt quantile function estimation method for censored samples.     

Estimation in Mixtures of Two Normal Distributions

This paper is concerned primarily with the method of moments in dissecting a mixture of two normal distributions. In the general case, with two means, two standard deviations, and a proportionality factor to be estimated, the first five sample moments are required, and it becomes necessary to find a particular solution of a ninth degree polynomial equation that was originally derived by Karl Pearson [10]. A procedure which circumvents solution of the nonic equation and thereby considerably reduces the total computational effort otherwise required, is presented. Estimates obtained in the simpler special case in which the two standard deviations are assumed to be equal, are employed as first approximations in an iterative method for simultaneously solving the basic system of moment equations applicable in the more general case in which the two standard deviations are unequal. Conditional maximum likelihood and conditional minimum chi-square estimation subject to having the first four sample moments equated to corresponding population moments, are also considered. An illustrative example is included.     

Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.     

On Identifying the Population of Origin of Each Observation in a Mixture of Observations from Two Gamma Populations

The gamma distribution, known also as the Erlangian distribution, and its special case the exponential distribution arise in many technological applications of statistics. The present note is on the problem of identifying the population of origin of each observation in a sample thought to be the result of mixing a random sample of size N ₁, from a gamma distribution with scale parameter σ₁ and an independent random sample of size N ₂ from another gamma distribution with scale parameter σ₂. We shall also be interested in the estimation of σ₁ and σ₂. The method of moments and the maximum likelihood method are applied to the solution of these problems.     

The beta exponential distribution

The exponential distribution is perhaps the most widely applied statistical distribution for problems in reliability. In this note, we introduce a generalization—referred to as the beta exponential distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the beta exponential distribution. We derive expressions for the moment generating function, characteristic function, the first four moments, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, the distribution of sums and ratios, and the asymptotic distribution of the extreme order statistics. We also discuss simulation issues, estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix. We hope that this generalization will attract wider applicability in reliability.     

Parameter and Quantile Estimation for the Generalized Pareto Distribution

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.     

An Approximation to the Negative Moments of the Positive Binomial Useful in Life Testing

The purpose of this paper is to obtain the mean and variance of the maximum likelihood estimator of the scale parameter of a Weibull distribution where the sample is censored at a fixed time. It will be shown that these moments are functions of the negative moments of the positive binomial distribution. A simple approximation is obtained for the negative moments of the positive binomial, thus giving an approximate expression for the mean and variance of the estimator.     

Nonparametric estimation for accelerated life tests under intermittent inspection

This paper considers nonparametric estimation of lifetime distribution based on grouped data from constant stress accelerated life tests under intermittent inspection in which test items are inspected only at specified points in time. A method of estimating the lifetime distribution at use condition stress is proposed for the case where the time transformation function relating stress to lifetime is a version of inverse power law. Numerical studies show that the proposed method is comparable to the maximum likelihood method for small sample size and is more accurate than existing nonparametric methods used for continuous inspection. The method performs better than the maximum likelihood method when the underlying lifetime distribution is incorrectly specified.     

Maximum Likelihood Estimation for the Multinomial Distribution Using Geometric Progamming

The problem of obtaining maximum likelihood estimates for the multinomial distribution is considered. Maximum likelihood estimation is applied to the particular problem of estimating overlap sizes created by interlocking sampling frames. In this paper, geometric programming is discusseda as a method of solving the likelihood function and is applied to a practical example of estimating overlap sizes.     

渤海北部冰厚重现值的极大似然法区间估计

不同重现期的年极值冰厚是有冰海区建筑物设计的关键指标。目前在确定设计冰厚时,往往只给出点估计。而求解设计冰厚的置信区间,在某一置信水平下获取设计值的范围,可以确定设计重现值的不确定性,给海工结构物的设计与建造提供指导。采用极大似然方法区间估计,给出了Gumbel分布、三参数Weibull分布、三参数对数正态分布和P-III型分布置信区间的求解过程。利用渤海北部营口和葫芦岛海区历年总冰厚极大值的实测数据,基于以上4种分布型式,采用极大似然法求得不同重现期下冰厚重现值的置信区间,并对各分布型式进行了优选比较。结果表明,选取P-III型分布求解这两地海冰厚度重现值的置信区间较优。     

On Modeling the Medical Care Insurance Data via a New Statistical Model

Proposing new statistical distributions which are more flexible than the existing distributions have become a recent trend in the practice of distribution theory. Actuaries often search for new and appropriate statistical models to address data related to financial and risk management problems. In the present study, an extension of the Lomax distribution is proposed via using the approach of the weighted T-X family of distributions. The mathematical properties along with the characterization of the new model via truncated moments are derived. The model parameters are estimated via a prominent approach called the maximum likelihood estimation method. A brief Monte Carlo simulation study to assess the performance of the model parameters is conducted. An application to medical care insurance data is provided to illustrate the potentials of the newly proposed extension of the Lomax distribution. The comparison of the proposed model is made with the (i) Two-parameter Lomax distribution, (ii) Three-parameter models called the half logistic Lomax and exponentiated Lomax distributions, and (iii) A four-parameter model called the Kumaraswamy Lomax distribution. The statistical analysis indicates that the proposed model performs better than the competitive models in analyzing data in financial and actuarial sciences.     

Maximum likelihood estimation techniques for concurrent flaw subpopulations

Failure of structural materials is often caused by the presence of two or more types of defect subpopulations. The maximum likelihood estimation technique for evaluating the Weibull parameters of these underlying subpopulations in the case of known fracture origin is presented. The maximum likelihood estimation equations are derived, and solved by means of nonlinear programming. The estimators obtained therefrom are tested for both accuracy and consistency against a series of simulation runs. For data sets containing a relatively small sample size, the advantage of the method of maximum likelihood over two established nonparametric techniques is demonstrated.     

基于广义高斯噪声模型的低频线谱检测

研究浅海环境噪声中低频线谱的检测问题.首先对实测海洋环境噪声的统计特性进行分析,提出了应用广义高斯分布对海洋环境噪声建模方法.在此基础上,针对低频线谱检测问题,推导了广义高斯噪声场中的最大似然检测器和渐近检测器,给出了弱线谱条件下线谱频率的最大似然估计.最后,利用实测数据对三种检测器的线谱检测性能进行了验证,结果表明:...     

Zubair Lomax Distribution: Properties and Estimation Based on Ranked Set Sampling

In this article, we offer a new adapted model with three parameters, called Zubair Lomax distribution. The new model can be very useful in analyzing and modeling real data and provides better fits than some others new models. Primary properties of the Zubair Lomax model are determined by moments, probability weighted moments, Renyi entropy, quantile function and stochastic ordering, among others. Maximum likelihood method is used to estimate the population parameters, owing to simple random sample and ranked set sampling schemes. The behavior of the maximum likelihood estimates for the model parameters is studied using Monte Carlo simulation. Criteria measures including biases, mean square errors and relative efficiencies are used to compare estimates. Regarding the simulation study, we observe that the estimates based on ranked set sampling are more efficient compared to the estimates based on simple random sample. The importance and flexibility of Zubair Lomax are validated empirically in modeling two types of lifetime data.