A robust estimator for the tail index of Pareto-type distributions

In extreme value statistics, the extreme value index is a well-known parameter to measure the tail heaviness of a distribution. Pareto-type distributions, with strictly positive extreme value index (or tail index) are considered. The most prominent extreme value methods are constructed on efficient maximum likelihood estimators based on specific parametric models which are fitted to excesses over large thresholds. Maximum likelihood estimators however are often not very robust, which makes them sensitive to few particular observations. Even in extreme value statistics, where the most extreme data usually receive most attention, this can constitute a serious problem. The problem is illustrated on a real data set from geopedology, in which a few abnormal soil measurements highly influence the estimates of the tail index. In order to overcome this problem, a robust estimator of the tail index is proposed, by combining a refinement of the Pareto approximation for the conditional distribution of relative excesses over a large threshold with an integrated squared error approach on partial density component estimation. It is shown that the influence function of this newly proposed estimator is bounded and through several simulations it is illustrated that it performs reasonably well at contaminated as well as uncontaminated data.     

A robust estimator for the tail index of Pareto-type distributions

In extreme value statistics, the extreme value index is a well-known parameter to measure the tail heaviness of a distribution. Pareto-type distributions, with strictly positive extreme value index (or tail index) are considered. The most prominent extreme value methods are constructed on efficient maximum likelihood estimators based on specific parametric models which are fitted to excesses over large thresholds. Maximum likelihood estimators however are often not very robust, which makes them sensitive to few particular observations. Even in extreme value statistics, where the most extreme data usually receive most attention, this can constitute a serious problem. The problem is illustrated on a real data set from geopedology, in which a few abnormal soil measurements highly influence the estimates of the tail index. In order to overcome this problem, a robust estimator of the tail index is proposed, by combining a refinement of the Pareto approximation for the conditional distribution of relative excesses over a large threshold with an integrated squared error approach on partial density component estimation. It is shown that the influence function of this newly proposed estimator is bounded and through several simulations it is illustrated that it performs reasonably well at contaminated as well as uncontaminated data.     

A quantile estimation for massive data with generalized Pareto distribution

This paper proposes a new method of estimating extreme quantiles of heavy-tailed distributions for massive data. The method utilizes the Peak Over Threshold (POT) method with generalized Pareto distribution (GPD) that is commonly used to estimate extreme quantiles and the parameter estimation of GPD using the empirical distribution function (EDF) and nonlinear least squares (NLS). We first estimate the parameters of GPD using EDF and NLS and then, estimate multiple high quantiles for massive data based on observations over a certain threshold value using the conventional POT. The simulation results demonstrate that our parameter estimation method has a smaller Mean square error (MSE) than other common methods when the shape parameter of GPD is at least 0. The estimated quantiles also show the best performance in terms of root MSE (RMSE) and absolute relative bias (ARB) for heavy-tailed distributions.     

Maximum likelihood parameter estimation algorithm for controlled autoregressive autoregressive models

A maximum likelihood parameter estimation algorithm is derived for controlled autoregressive autoregressive (CARAR) models based on the maximum likelihood principle. In this derivation, we use an estimated noise transfer function to filter the input–output data. The simulation results show that the proposed estimation algorithm can effectively estimate the parameters of such class of CARAR systems and give more accurate parameter estimates than the recursive generalized least-squares algorithm.     

Statistical Inferences for Generalized Pareto Distribution Based on Interior Penalty Function Algorithm and Bootstrap Methods and Applications in Analyzing Stock Data

This paper studies the application of extreme value statistics (EVS) theory on analysis for stock data, based on interior penalty function algorithm and Bootstrap methods. The generalized Pareto distribution (GPD) models are considered in analyzing the closing price data of Shanghai stock market. The maximum likelihood estimates (MLEs) are obtained by using the interior penalty function algorithm. Correspondingly, the bias and standard errors of MLEs, and the hypothesis test on the shape parameter are concerned through Bootstrap methods. Some simulations are performed to demonstrate the efficacy of parameter estimation and the power of the test. The estimates of the tail index in this paper are compared with those obtained via classical methods. At last, the model is diagnosed by numerical and graphical methods and the Value-at-Risk (VaR) is estimated.     

Confidence-based reliability assessment considering limited numbers of both input and output test data

Simulation-based methods can be used for accurate uncertainty quantification and prediction of the reliability of a physical system under the following assumptions: (1) accurate input distribution models and (2) accurate simulation models (including accurate surrogate models if utilized). However, in practical engineering applications, often only limited numbers of input test data are available for modeling input distribution models. Thus, estimated input distribution models are uncertain. In addition, the simulation model could be biased due to assumptions and idealizations used in the modeling process. Furthermore, only a limited number of physical output test data is available in the practical engineering applications. As a result, target output distributions, against which the simulation model can be validated, are uncertain and the corresponding reliabilities become uncertain as well. To assess the conservative reliability of the product properly under the uncertainties due to limited numbers of both input and output test data and a biased simulation model, a confidence-based reliability assessment method is developed in this paper. In the developed method, a hierarchical Bayesian model is formulated to obtain the uncertainty distribution of reliability. Then, we can specify a target confidence level. The reliability value at the target confidence level using the uncertainty distribution of reliability is the confidence-based reliability, which is the confidence-based estimation of the true reliability. It has been numerically demonstrated that the proposed method can predict the reliability of a physical system that satisfies the user-specified target confidence level, using limited numbers of input and output test data.     

Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.     

基于群智能的模糊多目标软件可靠性冗余分配

针对软件可靠性冗余分配问题,建立了一种模糊多目标分配模型,并提出了基于分布估计的细菌觅食优化算法求解该模型。将软件可靠性和成本作为模糊目标函数,通过三角形隶属函数对模糊多目标进行处理,用高斯分布对细菌觅食算法进行优化,并将该优化算法用来求解多目标软件可靠性冗余分配问题,设置不同的隶属函数参数可以得到不同的Pareto最优解,实验数据验证了该群智能算法对解决多目标软件可靠性分配的有效性和正确性,Pareto最优解可为在可靠性和成本之间决策提供依据。     

The beta generalized half-normal distribution

For the first time, we propose the so-called beta generalized half-normal distribution, which contains some important distributions as special cases, such as the half-normal and generalized half-normal (Cooray and Ananda, 2008) distributions. We derive expansions for the cumulative distribution and density functions which do not depend on complicated functions. We obtain formal expressions for the moments of the new distribution. We examine the maximum likelihood estimation of the parameters and provide the expected information matrix. The usefulness of the new distribution is illustrated through a real data set by showing that it is quite flexible in analyzing positive data instead of the generalized half-normal, half-normal, Weibull and beta Weibull distributions.     

Entropies of Mixing (EOM)and the Lorenz Order

Polynomial nonadditive, or pseudo-additive (PAE), entropies are related to the Shannon entropy in that both are derived from two classes of parent distributions of extreme-value theory, the Pareto and power distributions.The third class is the exponential distribution, corresponding to the Shannon entropy, to which the other two tend as their shape parameters increase without limit. These entropies all belong to a single class of entropies referred to as EOM. EOM is defined as the normalized difference between the dual of the Lorentz function and the Lorenz function. Sufficient conditions for majorization involve finding a separable, Schur-concave function, like the EOM, which increases as the distribution becomes more uniform or less spread out. Lorenz ordering has been associated to the degree in which the Lorenz curve is bent. This criterion is valid for tail distributions, and fails in the case where the distribution is limited on the right. EOM provide criteria for inequality in the Lorenz ordering sense: In the Pareto case,an increase in the shape parameter implies a decrease in inequality and the EOM decreases, whereas for the power distribution an increase in the shape parameter corresponds to an increase in inequality leading to an increase in the EOM. An analogy is drawn between Gauss' invariant distribution for the probability of the fractional part of a continued fraction and the area criterion in Lorenz ordering, analogous to the Gini index criterion. The tendency to approach the invariant distribution, as the number of partial quotients increases without limit, is shown to be analogous to the tendency to approach the invariant area, as the shape parameters increase without limit.     

Estimating the characteristics of randomized dynamic data models (the Entropy-Robust Approach)

We propose a new approach to finding dependencies between small volumes of input and output data based on randomized dynamic models and density estimation for the distributions of their parameters. Randomized dynamic models are defined by functional Volterra polynomials. To construct robust nonparametric estimation procedures, we develop an entropybased approach that employs functionals of generalized informational Boltzmann and Fermi entropies.     

基于Contourlet系数的广义高斯分布参数的混合估计方法

提出采用广义高斯概率密度建模的Contourlet变换系数的形状参数混合估计方法.当形状参数值较小时,对于小样本采用熵匹配方法估计,而对于大样本利用最大似然方法估计.当形状参数值较大时,采用矩方法估计.实验结果表明,所提出的方案可以有效准确地对Contourlet变换子带进行建模.     

A new exponential-type distribution with constant,decreasing, increasing,upside-down bathtub and bathtub-shaped failure rate function

A new three-parameter exponential-type family of distributions which can be used in modeling survival data, reliability problems and fatigue life studies is introduced. Its failure rate function can be constant, decreasing, increasing, upside-down bathtub or bathtub-shaped depending on its parameters. It includes as special sub-models the exponential distribution, the generalized exponential distribution [Gupta, R.D., Kundu, D., 1999. Generalized exponential distributions. Australian and New Zealand Journal of Statistics 41, 173–188] and the extended exponential distribution [Nadarajah, S., Haghighi, F., 2011. An extension of the exponential distribution. Statistics 45, 543–558]. A comprehensive account of the mathematical properties of the new family of distributions is provided. Maximum likelihood estimation of the unknown parameters of the new model for complete sample as well as for censored sample is discussed. Estimation of the stress–strength parameter is also considered. Two empirical applications of the new model to real data are presented for illustrative purposes.     

A global simulated annealing heuristic for the three-parameter lognormal maximum likelihood estimation

It is well known that the inclusion of the threshold parameter in a lognormal distribution creates serious complications for parameter estimation; several parameterized schemes and global optimization procedures have been proposed to solve the problem in the maximum likelihood framework. A global Simulated Annealing optimization heuristic is proposed to solve the problem of maximum likelihood estimation in any parameterization scheme for the three-parameter lognormal distribution, as well as for the extended lognormal distribution. Positively and negatively skewed lognormal distributions are considered by introducing a one-parameter conditional estimation procedure in the classical parameterization for the three-parameter lognormal distribution, and a dual reparameterization is introduced for parameters estimation in the extended lognormal distribution. Simulated and real data are analyzed to test the efficiency of the proposed algorithm.     

A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution

The Weibull distribution plays an important role in failure distribution modeling of reliability research. While there are three parameters in the general form of this distribution, for simplicity, one of its parameters is usually omitted and as a result, the others are estimated easily. However, due to its more flexibility, when the general form of the Weibull distribution is of interest, the estimation procedure is not an easy task anymore. For example, in the maximum likelihood estimation method, the likelihood function that is formed for a three-parameter Weibull distribution is very hard to maximize. In this paper, a new hybrid methodology based on a variable neighborhood search and a simulated annealing approach is proposed to maximize the likelihood function of a three-parameter Weibull distribution. The performance of the proposed methodology in terms of both the estimation accuracy and the required CPU time is then evaluated and compared to the ones of an existing current method through a wide range of numerical examples in which a sensitivity analysis is performed on the sample size. The results of the comparison study show that while the proposed method provides accurate estimates as well as those of the existing method, it requires significantly less CPU time.     

A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution

Three methods of estimation, namely maximum likelihood, moments and L-moments, when data come from an asymmetric exponential power distribution are considered. This is a very flexible four-parameter family exhibiting variety of tail and shape behaviours. The analytical expression of the first four L-moments of these distributions are derived, allowing for the use of L-moments estimators. A simulation study compares the three estimation methods in small samples.     

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.     

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.