Some Methods for Estimation in a Negative-Binomial Model

To clarify the advantage of using the quasilikelihood method, lack of robustness of the maximum likelihood method was demonstrated for the negative-binomial model. Efficiency calculations of the method of moments and the pseudolikelihood method in the estimation of extra-Poisson parameters in a negative-binomial model were carried out. Especially when the overdispersion parameter is small, both methods are relatively highly efficient and the pseudolikelihood estimate is more efficient than the method of moments estimate. Two examples of the quasilikelihood analyses of count data with overdispersion are given. The bootstrap method also is applied to the data to illustrate the advantage of the method of moments or pseudolikelihood method in the estimation of the standard errors of the mean parameter estimates under the negative-binomial model.     

Empirical processes with estimated parameters under auxiliary information

Empirical processes with estimated parameters are a well established subject in nonparametric statistics. In the classical theory they are based on the empirical distribution function which is the nonparametric maximum likelihood estimator for a completely unknown distribution function. In the presence of some “nonparametric” auxiliary information about the distribution, like a known mean or a known median, for example, the nonparametric maximum likelihood estimator is a modified empirical distribution function which puts random masses on the observations in order to take the available information into account [see Owen, Biometrika 75 (1988) 237–249, Ann. Statist. 18 (1990) 90–120, Empirical Likelihood, Chapman & Hall/CRC, London/Boca Raton, FL; Qin and Lawless, Ann. Statist. 22 (1994) 300–325]. Zhang [Metrika 46 (1997) 221–244] has proved a functional central limit theorem for the empirical process pertaining to this modified empirical distribution function. We will consider the corresponding empirical process with estimated parameters here and derive its asymptotic distribution. The limiting process is a centered Gaussian process with a complicated covariance function depending on the unknown parameter. The result becomes useful in practice through the bootstrap, which is shown to be consistent in case of a known mean. The performance of the resulting bootstrap goodness-of-fit test based on the Kolmogorov–Smirnov statistic is studied through simulations.     

Model selection for Gaussian latent block clustering with the integrated classification likelihood

Block clustering aims to reveal homogeneous block structures in a data table. Among the different approaches of block clustering, we consider here a model-based method: the Gaussian latent block model for continuous data which is an extension of the Gaussian mixture model for one-way clustering. For a given data table, several candidate models are usually examined, which differ for example in the number of clusters. Model selection then becomes a critical issue. To this end, we develop a criterion based on an approximation of the integrated classification likelihood for the Gaussian latent block model, and propose a Bayesian information criterion-like variant following the same pattern. We also propose a non-asymptotic exact criterion, thus circumventing the controversial definition of the asymptotic regime arising from the dual nature of the rows and columns in co-clustering. The experimental results show steady performances of these criteria for medium to large data tables.     

Bayesian Inference in Hidden Markov Random Fields for Binary Data Defined on Large Lattices

Hidden Markov random fields represent a complex hierarchical model, where the hidden latent process is an undirected graphical structure. Performing inference for such models is difficult primarily because the likelihood of the hidden states is often unavailable. The main contribution of this article is to present approximate methods to calculate the likelihood for large lattices based on exact methods for smaller lattices. We introduce approximate likelihood methods by relaxing some of the dependencies in the latent model, and also by extending tractable approximations to the likelihood, the so-called pseudolikelihood approximations, for a large lattice partitioned into smaller sublattices. Results are presented based on simulated data as well as inference for the temporal-spatial structure of the interaction between up- and down-regulated states within the mitochondrial chromosome of the Plasmodium falciparum organism. Supplemental material for this article is available online.     

Empirical Likelihood Ratio With Arbitrarily Censored/Truncated Data by EM Algorithm

The empirical likelihood is a general nonparametric inference procedure with many desirable properties. Recently, theoretical results for empirical likelihood with certain censored/truncated data have been developed. However, the computation of empirical likelihood ratios with censored/truncated data is often nontrivial. This article proposes a modified self-consistent/EM algorithm to compute a class of empirical likelihood ratios for arbitrarily censored/truncated data with a mean type constraint. Simulations show that the chi-square approximations of the log-empirical likelihood ratio perform well. Examples and simulations are given in the following cases: (1) right-censored data with a mean parameter; and (2) left-truncated and right-censored data with a mean type parameter.     

Reference Bayesian analysis of inverse Gaussian degradation process

In this paper, objective Bayesian method is applied to analyze degradation model based on the inverse Gaussian process. Noninformative priors (Jefferys prior and two reference priors) for model parameters are obtained and their properties are discussed. Moreover, we propose a class of modified reference priors to remedy weaknesses of the usual reference priors and show that the modified reference priors not only have proper posterior distributions but also have probability matching properties for model parameters. Gibbs sampling algorithms for Bayesian inference based on the Jefferys prior and the modified reference priors are studied. Simulations are conducted to compare the objective Bayesian estimates with the maximum likelihood estimates and subjective Bayesian estimates and shows better performance of the objective method than the other two estimates especially for the case of small sample size. Finally, two real data examples are analyzed for illustration.     

Computational Bayesian Analysis of Hidden Markov Models

Abstract

Versions of the Gibbs Sampler are derived for the analysis of data from hidden Markov chains and hidden Markov random fields. The principal new development is to use the pseudolikelihood function associated with the underlying Markov process in place of the likelihood, which is intractable in the case of a Markov random field, in the simulation step for the parameters in the Markov process. Theoretical aspects are discussed and a numerical study is reported.     

Geoadditive regression for analyzing small-scale geographical variability in car insurance

The empirical part of this article is based on a study on car insurance data to explore how global and local geographical effects on frequency and size of claims can be assessed with appropriate statistical methodology. Because these spatial effects have to be modeled and estimated simultaneously with linear and possibly nonlinear effects of further covariates such as age of policy holder, age of car or bonus-malus score, generalized linear models cannot be applied. Also, compared to previous analyses, the geographical information is given by the exact location of the residence of policy holders. Therefore, we employ a new class of geoadditive models, where the spatial component is modeled based on stationary Gaussian random fields, common in geostatistics (Kriging). Statistical inference is carried out by an empirical Bayes or penalized likelihood approach using mixed model technology. The results confirm that the methodological concept provides useful tools for exploratory analyses of the data at hand and in similar situations.     

Bootstrapping Pseudolikelihood Models for Clustered Binary Data

Asymptotic properties of the parametric bootstrap procedure for maximum pseudolikelihood estimators and hypothesis tests are studied in the general framework of associated populations. The technique is applied to the analysis of toxicological experiments which, based on pseudolikelihood inference for clustered binary data, fits into this framework. It is shown that the bootstrap approximation can be used as an interesting alternative to the classical asymptotic distribution of estimators and test statistics. Finite sample simulations for clustered binary data models confirm the asymptotic theory and indicate some substantial improvements.     

Estimation of Symmetry-Constrained Gaussian Graphical Models: Application to Clustered Dense Networks

We propose a model selection algorithm for high-dimensional clustered data. Our algorithm combines a classical penalized likelihood method with a composite likelihood approach in the framework of colored graphical Gaussian models. Our method is designed to identify high-dimensional dense networks with a large number of edges but sparse edge classes. Its empirical performance is demonstrated through simulation studies and a network analysis of a gene expression dataset.     

Empirical likelihood confidence intervals for hazard and density functions under right censorship

In this paper, we use smoothed empirical likelihood methods to construct confidence intervals for hazard and density functions under right censorship. Some empirical log-likelihood ratios for the hazard and density functions are obtained and their asymptotic limits are derived. Approximate confidence intervals based on these methods are constructed. Simulation studies are used to compare the empirical likelihood methods and the normal approximation methods in terms of coverage accuracy. It is found that the empirical likelihood methods provide better inference.     

非线性回归模型的经验似然诊断

经验似然方法已经被广泛用于线性模型和广义线性模型.本文基于经验似然方法对非线性回归模型进行统计诊断.首先得到模型参数的极大经验似然估计;其次基于经验似然研究了三种不同的影响曲率度量;最后通过一个实际例子,说明了诊断方法的有效性.     

Asymptotic tests of composite hypotheses for non-ergodic type stochastic processes

Limiting distributions of a score statistic and the likelihood ratio statistic for testing a composite hypothesis involving several parameters in non-ergodic type stochastic processes are obtained. It is shown that, unlike in the usual theory (ergodic type processes), the limiting distributions of these statistics are different both under the null and a contiguous sequence of alternative hypotheses. The results are applied to a regression model with explosive autoregressive Gaussian errors. In the discussion of this example a modified score statistic is suggested where the limiting null and non-null distributions are the same as those of the likelihood ratio statistic.     

Quantile Processes in the Presence of Auxiliary Information

We employ the empirical likelihood method to propose a modified quantile process under a nonparametric model in which we have some auxiliary information about the population distribution. Furthermore, we propose a modified bootstrap method for estimating the sampling distribution of the modified quantile process. To explore the asymptotic behavior of the modified quantile process and to justify the bootstrapping of this process, we establish the weak convergence of the modified quantile process to a Gaussian process and the almost-sure weak convergence of the modified bootstrapped quantile process to the same Gaussian process. These results are demonstrated to be applicable, in the presence of auxiliary information, to the construction of asymptotic bootstrap confidence bands for the quantile function. Moreover, we consider estimating the population semi-interquartile range on the basis of the modified quantile process. Results from a simulation study assessing the finite-sample performance of the proposed semi-interquartile range estimator are included.     

基于经验似然的部分线性模型的统计诊断

经验似然方法己经被广泛应用于许多模型的统计推断.本文基于经验似然对部分线性模型进行统计诊断.首先给出模型的估计方程,进而得到模型参数的极大经验似然估计;其次,基于经验似然研究了三种不同的影响曲率;最后通过随机模拟和实例分析,说明了统计诊断方法的有效性.     

深圳成分指数与上海综合指数的稳定分布

本文利用最大似然估计法与特征变换法分别对深圳成分指数(SZSI)与上海综合指数(SHCl)的日收益率分布进行了稳定分布的拟合,并进行了比较,验证了SZSI与SHCI日收益率序列的尖峰厚尾现象。     

Bivariate Distributions with Pearson Type VII Conditionals

In this article the most general class of bivariate distributions such that both conditional densities are Pearson Type VII, with fixed shape parameter, is fully characterized. Some of its properties and relations with other distributions are explored. The estimation of parameters is considered by the methods of maximum likelihood and pseudolikelihood and a method for random variate generation is presented along with a simulation experiment. Bivariate and multivariate extensions of the Pearson Type VII conditionals distribution are also discussed.     

缺失数据均值推断的经验似然方法

研究了缺失数据的均值推断问题.在随机缺失及半参数模型的假设下,设计了基于影响函数理论的经验似然推断方法,证明了所构造的对数经验似然比检验统计量具有非参数Wilks性质.此外,该经验似然方法可以利用辅助协变量中提供的附加信息来提高检验的功效.在近邻备择假设下,计算了检验统计量的功效,并且通过一些模拟考察了该方法在有限样本下的表现.     

Jackknife-blockwise empirical likelihood methods under dependence

Empirical likelihood for general estimating equations is a method for testing hypothesis or constructing confidence regions on parameters of interest. If the number of parameters of interest is smaller than that of estimating equations, a profile empirical likelihood has to be employed. In case of dependent data, a profile blockwise empirical likelihood method can be used. However, if too many nuisance parameters are involved, a computational difficulty in optimizing the profile empirical likelihood arises. Recently, Li et al. (2011) [9] proposed a jackknife empirical likelihood method to reduce the computation in the profile empirical likelihood methods for independent data. In this paper, we propose a jackknife-blockwise empirical likelihood method to overcome the computational burden in the profile blockwise empirical likelihood method for weakly dependent data.     

Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform

This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.