Parameter estimation of the generalized gamma distribution

This article focuses on the parameter estimation of the generalized gamma distribution. Because of many difficulties described in the literature to estimate the parameters, we propose here a new estimation method. The algorithm associated to this heuristic method is implemented in Splus. We validate the resulting routine on the particular cases of the generalized gamma distribution.     

An alternative inverse Gaussian distribution

An alternative inverse Gaussian distribution expressed in terms of the Bessel function is introduced. Both theoretical and empirical motivation is provided. Various particular cases and expressions for moments are derived. Estimation procedures by the method of moments and the method of maximum likelihood as well as the associated Fisher information matrix are derived. A simulation study is performed to investigate the asymptotic distribution of an associated boundary crossing variable. Finally, an application is illustrated to show that the proposed distribution can be a better model for reliability data than one based on the standard inverse Gaussian distribution.     

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.     

A convenient way of generating gamma random variables using generalized exponential distribution

In this paper we propose a very convenient way to generate gamma random variables using generalized exponential distribution, when the shape parameter lies between 0 and 1. The new method is compared with the most popular Ahrens and Dieter method and the method proposed by Best. Like Ahrens and Dieter and Best methods our method also uses the acceptance-rejection principle. But it is observed that our method has greater acceptance proportion than Ahrens and Dieter or Best methods.     

Mixture and non-mixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data

The cure fraction models are usually used to model lifetime time data with long-term survivors. In the present article, we introduce a Bayesian analysis of the four-parameter generalized modified Weibull (GMW) distribution in presence of cure fraction, censored data and covariates. In order to include the proportion of “cured” patients, mixture and non-mixture formulation models are considered. To demonstrate the ability of using this model in the analysis of real data, we consider an application to data from patients with gastric adenocarcinoma. Inferences are obtained by using MCMC (Markov Chain Monte Carlo) methods.     

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.     

The beta Burr XII distribution with application to lifetime data

For the first time, a five-parameter distribution, the so-called beta Burr XII distribution, is defined and investigated. The new distribution contains as special sub-models some well-known distributions discussed in the literature, such as the logistic, Weibull and Burr XII distributions, among several others. We derive its moment generating function. We obtain, as a special case, the moment generating function of the Burr XII distribution, which seems to be a new result. Moments, mean deviations, Bonferroni and Lorenz curves and reliability are provided. We derive two representations for the moments of the order statistics. The method of maximum likelihood and a Bayesian analysis are proposed for estimating the model parameters. The observed information matrix is obtained. For different parameter settings and sample sizes, various simulation studies are performed and compared in order to study the performance of the new distribution. An application to real data demonstrates that the new distribution can provide a better fit than other classical models. We hope that this generalization may attract wider applications in reliability, biology and lifetime data analysis.     

适应交通流演化的伽马分布形状参数估计

研究交通流演化规律对交通安全措施的实施具有重要意义。交通流的演化过程一般可分为交通流的量变与质变。鉴于交通自由流状态、拥挤流状态及间歇流状态的分布均可由伽马分布表达,利用伽马分布及其变点检验算法进行变点搜索及检验,确定变点前后的交通流状态。结合柯尔莫哥洛夫-斯米尔诺夫检验方法对观测数据是否服从伽马分布进行拟合优度检验。为较直观说明交通流的演化规律,结合极大似然估计,给出了适应交通流演化的伽马分布形状参数估计。实验结果表明,伽马分布形状参数估计的研究是探索交通流演化规律的有利途径。     

Random weighting estimation of parameters in generalized Gaussian distribution

This paper studies random weighting estimation of shape and scale parameters in generalized Gaussian distribution (GGD). An expression is established to describe the relationship between moments and parameters. The strong convergence for random weighting estimation of GGD parameters is also rigorously proved. Computational simulations and practical experiments are presented to demonstrate the efficacy for random weighting estimation of GGD parameters.     

From data to global generalized knowledge

The attribute-oriented induction (AOI) is a useful data mining method that extracts generalized knowledge from relational data and user's background knowledge. The method uses two thresholds, the relation threshold and attribute threshold, to guide the generalization process, and output generalized knowledge, a set of generalized tuples which describes the major characteristics of the target relation. Although AOI has been widely used in various applications, a potential weakness of this method is that it only provides a snapshot of the generalized knowledge, not a global picture. When thresholds are different, we would obtain different sets of generalized tuples, which also describe the major characteristics of the target relation. If a user wants to ascertain a global picture of induction, he or she must try different thresholds repeatedly. That is time-consuming and tedious. In this study, we propose a global AOI (GAOI) method, which employs the multiple-level mining technique with multiple minimum supports to generate all interesting generalized knowledge at one time. Experiment results on real-life dataset show that the proposed method is effective in finding global generalized knowledge.     

L-moments and TL-moments of the generalized lambda distribution

The 4-parameter generalized lambda distribution (GLD) is a flexible distribution capable of mimicking the shapes of many distributions and data samples including those with heavy tails. The method of L-moments and the recently developed method of trimmed L-moments (TL-moments) are attractive techniques for parameter estimation for heavy-tailed distributions for which the L- and TL-moments have been defined. Analytical solutions for the first five L- and TL-moments in terms of GLD parameters are derived. Unfortunately, numerical methods are needed to compute the parameters from the L- or TL-moments. Algorithms are suggested for parameter estimation. Application of the GLD using both L- and TL-moment parameter estimates from example data is demonstrated, and comparison of the L-moment fit of the 4-parameter kappa distribution is made. A small simulation study of the 98th percentile (far-right tail) is conducted for a heavy-tail GLD with high-outlier contamination. The simulations show, with respect to estimation of the 98th-percent quantile, that TL-moments are less biased (more robost) in the presence of high-outlier contamination. However, the robustness comes at the expense of considerably more sampling variability.     

Local likelihood regression in generalized linear single-index models with applications to microarray data

Searching for an effective dimension reduction space is an important problem in regression, especially for high-dimensional data such as microarray data. A major characteristic of microarray data consists in the small number of observations n and a very large number of genes p. This “large p, small n” paradigm makes the discriminant analysis for classification difficult. In order to offset this dimensionality problem a solution consists in reducing the dimension. Supervised classification is understood as a regression problem with a small number of observations and a large number of covariates. A new approach for dimension reduction is proposed. This is based on a semi-parametric approach which uses local likelihood estimates for single-index generalized linear models. The asymptotic properties of this procedure are considered and its asymptotic performances are illustrated by simulations. Applications of this method when applied to binary and multiclass classification of the three real data sets Colon, Leukemia and SRBCT are presented.     

Image thresholding based on the EM algorithm and the generalized Gaussian distribution

In this paper, a novel parametric and global image histogram thresholding method is presented. It is based on the estimation of the statistical parameters of “object” and “background” classes by the expectation-maximization (EM) algorithm, under the assumption that these two classes follow a generalized Gaussian (GG) distribution. The adoption of such a statistical model as an alternative to the more common Gaussian model is motivated by its attractive capability to approximate a broad variety of statistical behaviors with a small number of parameters. Since the quality of the solution provided by the iterative EM algorithm is strongly affected by initial conditions (which, if inappropriately set, may lead to unreliable estimation), a robust initialization strategy based on genetic algorithms (GAs) is proposed. Experimental results obtained on simulated and real images confirm the effectiveness of the proposed method.     

A simple method for representing some univariate frequency distributions, with particular application in Monte Carlo-based simulation

Introduced in this paper is a simple yet effective method for representing some of the types of univariate frequency distribution that commonly are required in Monte Carlo-based simulation. To use the method, an appropriate parent distribution is first chosen; then this distribution is modified by blending a constant value into the density function; the particular value used is the ordinate of the density function at its mode. The advantages of the method are (1) that a wide variety of forms of distribution can be represented, (2) that the number of parameters is low, (3) that the parameters can be varied continuously to let sets of systematically related distributions be constructed, and (4) that the resulting distribution functions are straightforward to invert numerically, thereby letting random deviates be generated quickly and efficiently. The method is potentially of particular value in Monte Carlo-based simulation, because it allows distributions of greatly differing forms to be represented within a single, flexible framework. The paper describes the method, provides the necessary equations for parameter estimation, and gives an example of a simulation exercise in which the method proved valuable. A demonstration program is provided that allows experimentation with the method.     

A mixture of generalized latent variable models for mixed mode and heterogeneous data

In the behavioral, biomedical, and social-psychological sciences, mixed data types such as continuous, ordinal, count, and nominal are common. Subpopulations also often exist and contribute to heterogeneity in the data. In this paper, we propose a mixture of generalized latent variable models (GLVMs) to handle mixed types of heterogeneous data. Different link functions are specified to model data of multiple types. A Bayesian approach, together with the Markov chain Monte Carlo (MCMC) method, is used to conduct the analysis. A modified DIC is used for model selection of mixture components in the GLVMs. A simulation study shows that our proposed methodology performs satisfactorily. An application of mixture GLVM to a data set from the National Longitudinal Surveys of Youth (NLSY) is presented.     

A regularity model-based multiobjective estimation of distribution algorithm with reducing redundant cluster operator

A regularity model-based multiobjective estimation of distribution algorithm (RM-MEDA) has been proposed for solving continuous multiobjective optimization problems with variable linkages. RM-MEDA is a kind of estimation of distribution algorithms and, therefore, modeling plays a critical role. In RM-MEDA, the population is split into several clusters to build the model. Moreover, the fixed number of clusters is recommended in RM-MEDA when solving different kinds of problems. However, based on our experiments, we find that the number of clusters is problem-dependent and has a significant effect on the performance of RM-MEDA. Motivated by the above observation, in this paper we improve the clustering process and propose a reducing redundant cluster operator (RRCO) to build more precise model during the evolution. By combining RRCO with RM-MEDA, we present an improved version of RM-MEDA, named IRM-MEDA. In this paper, we also construct four additional continuous multiobjective optimization test instances. The experimental results have shown that IRM-MEDA outperforms RM-MEDA in terms of efficiency and effectiveness. In particular, IRM-MEDA performs on average 31.67% faster than RM-MEDA.     

Bounded generalized Gaussian mixture model

The generalized Gaussian mixture model (GGMM) provides a flexible and suitable tool for many computer vision and pattern recognition problems. However, generalized Gaussian distribution is unbounded. In many applications, the observed data are digitalized and have bounded support. A new bounded generalized Gaussian mixture model (BGGMM), which includes the Gaussian mixture model (GMM), Laplace mixture model (LMM), and GGMM as special cases, is presented in this paper. We propose an extension of the generalized Gaussian distribution in this paper. This new distribution has a flexibility to fit different shapes of observed data such as non-Gaussian and bounded support data. In order to estimate the model parameters, we propose an alternate approach to minimize the higher bound on the data negative log-likelihood function. We quantify the performance of the BGGMM with simulations and real data.     

A systematic approach to determining metamodel scope for risk-based optimization and its application to water distribution system design

Metamodels have proven be very useful when it comes to reducing the computational requirements of Evolutionary Algorithm-based optimization by acting as quick-solving surrogates for slow-solving fitness functions. The relationship between metamodel scope and objective function varies between applications, that is, in some cases the metamodel acts as a surrogate for the whole fitness function, whereas in other cases it replaces only a component of the fitness function. This paper presents a formalized qualitative process to evaluate a fitness function to determine the most suitable metamodel scope so as to increase the likelihood of calibrating a high-fidelity metamodel and hence obtain good optimization results in a reasonable amount of time. The process is applied to the risk-based optimization of water distribution systems; a very computationally-intensive problem for real-world systems. The process is validated with a simple case study (modified New York Tunnels) and the power of metamodelling is demonstrated on a real-world case study (Pacific City) with a computational speed-up of several orders of magnitude.     

采用两步训练法的多目标分布估计算法

提出两步训练法,改进了基于规则模型的多目标分布估计算法.在算法的模型训练环节,首先采用均值分簇法进行初步聚类;然后采用基于流形分簇法进行细致聚类,代替原算法中采用局部主元分析算法需要循环迭代的聚类分簇方法.通过6个Benchmark测试函数验证,改进算法保持了原算法的收敛性和多样性,并缩短了寻优的时间.