广义Gamma分布簇广义线性混合模型的参数估计

文章在广义线性混合模型的框架下分析广义Gamma分布簇广义线性混合模型的参数估计问题.提出的参数估计有以下三个显著特点:其一,充分利用广义线性混合模型参数估计的理论成果,参数估计表达式与6种广义线性混合模型参数估计结果形式相同;其二,该方法可以通过反复调用广义线性混合模型中的参数估计模块来实现;其三,具有渐进正态性.     

广义线性混合模型框架下的信度模型分析

尝试在广义线性混合模型的框架下构建信度模型。在广义线性混合模型框架中,假定被解释变量服从指数簇分布,假定自然参数先验分布为相应的自然共轭先验分布簇,按照Bayes理论,通过特殊构造,给出推论:对随机效应的估计满足经典信度公式。参数估计部分,利用自然共轭先验分布簇参数子列上下极限的性质找出先验分布参数的含义和关系,使用伪似然方法给出信度估计公式。并以特例形式讨论Tweedie模型,对模型进行变形,得到特例的Bühlmann-Straub信度和经典的Bühlmann信度。该模型同时考虑先验信息与后验信息,对整合分类费率与个体经验费率提供一定参考。     

广义线性混合模型及其SAS实现

本文探讨了重复测量资料广义线性混合模型（GLMMs）建模及SAS9．1的PROC GLIMMIX程序实现。利用PROC GLIMMIX程序中Model语句选项和Link语句来指定因变量的分布及连接函数，通过Random语句来指定随机效应，采用线性限制性／残差虚拟似然法进行参数估计。GLMMs是在广义线性模型的基础上引入随机效应,因变量可以是指数家族中任意分布，可以通过连接函数将观测的均数向量与模型参数联系起来。GLMMs应用范围广，建模灵活，可以为相关或非常数方差数据建模，能提供客观正确的统计结论。     

混合广义线性模型

本文首次提出混合广义线性模型，此模型包括通常的标准广义线性模型以及贝叶斯多层广义线性模型。对二分量混合广义线性模型，利用近似准似然方法讨论其参数估计。对指数族混合广义线性模型，利用标准的广义线性模型分析方法得到参数的迭代估计。至于Ａｌｂｅｒｔ提出的贝叶斯多层先验分析方法，我们给出简单的讨论与修正；并且讨论与分析两个特殊的贝叶斯多层广义线性模型，给出它们的有关详细结果。最后，对混合广义线性模型，提出两个问题     

联合广义线性模型中的变量选择

在联合广义线性模型中,散度参数与均值都被赋予了广义线性模型的结构,本文主要考虑在只有分布的一阶矩和二阶矩指定的条件下,联合广义线性模型中均值部分的变量选择问题。本文采用广义拟似然函数,提出了新的模型选择准则(EAIC);该准则是Akaike信息准则的推广。论文通过模拟研究验证了该准则的效果。     

广义线性模型的性质及其在非寿险中的应用

广义线性模型的误差项服从指数分布族，通常的指数分布族包括正态分布、泊松分布、二项分布、伽玛分布、逆高斯分布等，这些分布模型在非寿险精算中都有广泛的应用。在对上述常见模型特点分析的同时，用实际数据进行了拟合，为精算师在实务工作中提供了些建议。     

广义线性混合模型及其在非寿险信度费率厘定中的应用

近年来,国内外精算学者开始将广义线性混合模型用于信度模型费率厘定中,但他们对因变量的推广仅仅推广到负二项分布。在前人的研究基础上,将因变量进一步推广到负二项K、广义泊松、双泊松等分布,然后用极大似然估计中的限制性虚拟似然法和自适应高斯求积法对参数进行估计,最后用美国劳工补偿保险进行实证分析。结果表明:负二项K(K=1.947)广义线性混合模型对数据拟合效果最好,其次为负二项1、负二项2、双泊松、广义泊松和泊松广义线性混合模型。     

广义线性模型下ERIC方法的调节参数选择

变量选择是处理高维统计模型的基本方法,在回归模型的变量选择中SCAD惩罚函数不仅可以很好地选择出正确模型,同时还可以对参数进行估计,而且还具有oracle性质,但这些良好的性质是基于选择出一个合适的调节参数。目前国内关于调节参数选择方面大多是对于变量选择问题的研究,针对广义线性模型基于SCAD惩罚使用新方法 ERIC准则进行调节参数的选择,并证明在一定条件下经过该准则选择的模型具有一致性。模拟与实证分析结果表明,ERIC方法在选择调节参数方面优于传统的CV准则、AIC准则和BIC准则。     

广义线性模型在非寿险精算中的应用及其研究进展

广义线性模型在精算中的应用始于20世纪80年代，其应用涉及到精算学的各个领域，如生命表的修匀、损失分布、信度理论、风险分类、准备金和费率估计等方面。在对广义线性模型适用于非寿险精算的典型特征进行分析的基础上，对广义线性模型在非寿险精算中的应用及其研究进展进行分析和总结的同时，重点分析利率厘定和准备金估计中广义线性模型的建模思想，并结合实际提出了今后研究的方向。     

Monte Carlo EM梯度法在MATLAB中的实现

文章使用空间广义线性混合模型为连续空间非正态变量建模,在MATLAB中实现模型参数估计的MCEMG算法,即结合Monte Carlo样本的EM梯度法,求解参数的极大似然估计及采样点随机效应的最小均方误估计。在GS+中进行随机效应的普通克里格插值,并最终对非采样点响应变量进行预测。模拟仿真结果显示该方法参数估计与真实值较接近,响应变量预测结果能反应真实数据总体分布情况。     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.     

Comparison of methods for analyzing binary repeated measures data: A simulation-based study (comparison of methods for binary repeated measures)

In this study, some methods suggested for binary repeated measures, namely, Weighted Least Squares (WLS), Generalized Estimating Equations (GEE), and Generalized Linear Mixed Models (GLMM) are compared with respect to power, type 1 error, and properties of estimates. The results indicate that with adequate sample size, no missing data, the only covariate being time effect, and a relatively limited number of time points, the WLS method performs well. The GEE approach performs well only for large sample sizes. The GLMM method is satisfactory with respect to type I error, but its estimates have poorer properties than the other methods.     

Modeling healthcare costs in simultaneous presence of asymmetry,heteroscedasticity and correlation

Highly skewed outcome distributions observed across clusters are common in medical research. The aim of this paper is to understand how regression models widely used for accommodating asymmetry fit clustered data under heteroscedasticity. In a simulation study, we provide evidence on the performance of the Gamma Generalized Linear Mixed Model (GLMM) and log-Linear Mixed-Effect (LME) model under a variety of data-generating mechanisms. Two case studies from health expenditures literature, the cost of strategies after myocardial infarction randomized clinical trial on the cost of strategies after myocardial infarction and the European Pressure Ulcer Advisory Panel hospital prevalence survey of pressure ulcers, are analyzed and discussed. According to simulation results, the log-LME model for a Gamma response can lead to estimations that are biased by as much as 10% of the true value, depending on the error variance. In the Gamma GLMM, the bias never exceeds 1%, regardless of the extent of heteroscedasticity, and the confidence intervals perform as nominally stated under most conditions. The Gamma GLMM with a log link seems to be more robust to both Gamma and log-normal generating mechanisms than the log-LME model.     

Fitting Generalized Hyperbolic processes - New insights for generating initial values

The fitting of Lévy processes is an important field of interest in both option pricing and risk management. In literature, a large number of fitting methods requiring adequate initial values at the start of the optimization procedure exists. A so-called simplified method of moments (SMoM) generates by assuming a symmetric distribution these initial values for the Variance Gamma process, whereby the idea behind can be easily transferred to the Normal Inverse Gaussian process. However, the characteristics of the Generalized Hyperbolic process prevent such an easy adaption. Therefore, we provide by applying a Taylor series approximation for the modified Bessel function of the third kind, a Tschirnhaus transformation and a symmetric distribution assumption, a SMOM for the Generalized Hyperbolic distribution. Our simulation study compares the results of our SMoM with the results of the maximum likelihood estimation. The results show that our proposed approach is an appropriate and useful way for estimating Generalized Hyperbolic process parameters and significantly reduces estimation time.     

Identification of parametric Rasch-type models

In modern Item Response Theory, the Rasch model is viewed as a Generalized Linear Mixed Model, where the item parameters correspond to the fixed-effects, whereas the person specific parameters are the random-effects. The statistical model, bearing on the observable variables only, is obtained after integrating out the random-effects. Although it is widely accepted that the parameters of this model are identified, it is hard to find a correct justification. Furthermore, the meaning of the parameters of the Rasch model – as well as of its extensions – is typically based on the fixed-effects specification of the model, that is, when the person specific parameters are also treated as fixed-effects. The contribution of this paper is to provide an explicit proof of the identification of the random-effects Rasch model. The proof is valid for a large class of Rasch-type models. It is also shown that such a proof can be applied to analyze the identification of Explanatory Rasch Models. Finally, the meaning of the parameters of interest with respect to the different data generating process is discussed.     

Generalized Systematic Sampling

In this paper, we introduce a new systematic sampling design, called a Generalized Systematic Sampling (GSS), for estimation of finite population mean. The proposed design is found to be better than Simple Random Sampling (SRS) and the generalization of the several existing systematic sampling schemes such as, Linear Systematic Sampling (LSS), Diagonal Systematic Sampling (DSS), and Generalized Diagonal Systematic Sampling (GDSS). All of these designs become special cases of the proposed design.     

Model Checks for Generalized Linear Models

In this paper we propose and study non-parametric tests for the validity of (composite) Generalized Linear Models with a given parametric link structure, which are based on certain empirical processes marked by the residuals. When properly transformed to their innovation part the resulting test statistics are distribution-free. The method perfectly adapts to a situation, when also the input vector follows a dimension reducing model.     

Conditions for D A -Maximin Marginal Designs for Generalized Linear Mixed Models to be Uniform

Optimal design theory deals with the assessment of the optimal joint distribution of all independent variables prior to data collection. In many practical situations, however, covariates are involved for which the distribution is not previously determined. The optimal design problem may then be reformulated in terms of finding the optimal marginal distribution for a specific set of variables. In general, the optimal solution may depend on the unknown (conditional) distribution of the covariates. This article discusses the D _A -maximin procedure to account for the uncertain distribution of the covariates. Sufficient conditions will be given under which the uniform design of a subset of independent discrete variables is D _A -maximin. The sufficient conditions are formulated for Generalized Linear Mixed Models with an arbitrary number of quantitative and qualitative independent variables and random effects.     

On the Limit of the Generalized Binomial Distribution

ABSTRACT

In this article we investigate the limit of the Generalized Binomial Distribution that results from correlated Bernoulli processes as the number of trials goes to infinity. For a correlation factor less than or equal to one half, we show, under certain assumptions, that the limit distribution is the standardized normal. For a correlation factor greater than one half, we find empirically that the limit is different from the standardized normal distribution.