Jackknifing in generalized linear models

In a generalized linear model, the jackknife estimator of the asymptotic covariance matrix of the maximum likelihood estimator is shown to be consistent. The corresponding jackknife studentized statistic is asymptotically normal. In addition, these results remain true even if there exist unequal dispersion parameters in the model. On the other hand, the variance estimator and the studentized statistic based on the standard method (substitution and linearization) do not enjoy this robustness property against the presence of unequal dispersion parameters.This research was supported by an Operating Grant from the Natural Science and Engineering Research Council of Canada.     

基于对数正态分布下联合均值与散度广义线性模型的极大似然估计

基于对数正态分布研究提出了联合均值与散度广义线性模型,给出了此模型参数的极大似然估计,模拟和实例显示该模型和方法是有用和有效的.     

双重广义线性模型的经验似然推断

基于截面经验似然方法,将双重广义线性模型的拟似然估计方程作为截面经验似然比函数的约束条件,构造了均值模型和散度模型未知参数的置信区间.最后通过数据模拟,将该方法与正态逼近方法比较,说明了该方法是有效和可行的.     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

The generalized linear exponential distribution

This paper deals with a new generalization of the linear exponential distribution. This distribution is called the generalized linear exponential distribution (GLED). Some statistical properties such as moments, modes and quantiles are derived. The failure rate function and the mean residual lifetime are also discussed. The maximum likelihood estimators of the parameters are obtained using a simulation study. Real data are used to determine whether the GLED is better than other well-known distributions in modeling lifetime data or not.     

Exponentiated generalized linear exponential distribution

A new generalization of the linear exponential distribution is recently proposed by Mahmoud and Alam [1], called as the generalized linear exponential distribution. Another generalization of the linear exponential was introduced by Sarhan and Kundu  and , named as the generalized linear failure rate distribution. This paper proposes a more generalization of the linear exponential distribution which generalizes the two. We refer to this new generalization as the exponentiated generalized linear exponential distribution. The new distribution is important since it contains as special sub-models some widely well known distributions in addition to the above two models, such as the exponentiated Weibull distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Three real data sets are analyzed using the new distribution, which show that the exponentiated generalized linear exponential distribution can be used quite effectively in analyzing real lifetime data.     

Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

Adaptive quasi-likelihood estimate in generalized linear models

This paper gives a thorough theoretical treatment on the adaptive quasi-likelihood estimate of the parameters in the generalized linear models. The unknown covariance matrix of the response variable is estimated by the sample. It is shown that the adaptive estimator defined in this paper is asymptotically most efficient in the sense that it is asymptotic normal, and the covariance matrix of the limit distribution coincides with the one for the quasi-likelihood estimator for the case that the covariance matrix of the response variable is completely known.     

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution

Starting from the question: What is the accident risk of an insured individual?, we consider that the customer has contracted policies in different insurance lines: motor and home. Three models based on the multivariate Sarmanov distribution are analyzed. Driven by a real data set that takes into account three types of accident risks, two for motor and one for home, three trivariate Sarmanov distributions with generalized linear models (GLMs) for marginals are considered and fitted to the data. To estimate the parameters of these three models, we discuss a method for approaching the maximum likelihood (ML) estimators. Finally, the three models are compared numerically with the simpler trivariate Negative Binomial GLM and with elliptical copula based models.     

Similar tests for covariance structures in multivariate linear models

Nyblom (J. Multivariate Anal. 76 (2001) 294) has derived locally best invariant test for the covariance structure in a multivariate linear model. The class of invariant tests obtained by Nyblom [9] does not coincide with the class of similar tests for this testing set-up. This paper extends some of the results of Nyblom [9] by deriving the locally best similar tests for the covariance structure. Moreover, it develops a saddlepoint approximation to optimal weighted average power similar tests (i.e. tests which maximize a weighted average power).     

Prediction and calibration in generalized linear models

This paper concerns prediction and calibration in generalized linear models. A new predictive procedure, giving improved prediction intervals, is briefly reviewed and further theoretical results, useful for calculations, are presented. Indeed, the calibration problem is faced within the classical approach and a suitable solution is obtained by inverting the associated improved prediction procedure. This calibration technique gives accurate confidence regions and it constitutes a substantial improvement over both the estimative solution and the naive solution, which involves, even for non-linear and non-normal models, the results available for the linear Gaussian case. Finally, some useful explicit formulae for the construction of prediction and calibration intervals are presented, with regard to generalized linear models with alternative error terms and link functions. This research was partially supported by a grant from Ministero dell’Instruzione, dell’Università e della Ricerca, Italy.     

Fuzzy clusterwise quasi-likelihood generalized linear models

The quasi-likelihood method has emerged as a useful approach to the parameter estimation of generalized linear models (GLM) in circumstances where there is insufficient distributional information to construct a likelihood function. Despite its flexibility, the quasi-likelihood approach to GLM is currently designed for an aggregate-sample analysis based on the assumption that the entire sample of observations is taken from a single homogenous population. Thus, this approach may not be suitable when heterogeneous subgroups exist in the population, which involve qualitatively distinct effects of covariates on the response variable. In this paper, the quasi-likelihood GLM approach is generalized to a fuzzy clustering framework which explicitly accounts for such cluster-level heterogeneity. A simple iterative estimation algorithm is presented to optimize the regularized fuzzy clustering criterion of the proposed method. The performance of the proposed method in recovering parameters is investigated based on a Monte Carlo analysis involving synthetic data. Finally, the empirical usefulness of the proposed method is illustrated through an application to actual data on the coupon usage behaviour of a sample of consumers.     

半参数广义线性模型的影响分析与异常点检验

统计诊断就是对统计推断方法解决问题的全过程进行诊断，而影响分析是统计诊断中十分重要的分支．本文针对半参数广义线性模型，证明了数据删除模型和均值漂移模型的等价性定理，给出了诸如广义Cook距离等诊断统计量并研究了异常点的Score检验统计量，最后通过实例验证了本文给出的诊断方法的有效性。     

Adaptive Lasso estimators for ultrahigh dimensional generalized linear models

Generalized linear models have been more widely used than linear models which exclude categorical variables. The penalized method becomes an effective tool to study ultrahigh dimensional generalized linear models. In this paper, we study theoretical results of the adaptive Lasso for generalized linear models in terms of diverging number of parameters and ultrahigh dimensionality. The asymptotic results are examined by several simulation studies.     

混合双重广义线性模型的参数估计

在实际应用中,不同类别的数据统计特性存在差异,所以对异质总体的研究非常有必要.基于总体一,二阶矩存在,利用双重广义线性模型对异质总体的不同子类数据的均值和散度同时建模,研究提出了混合双重广义线性模型.然后,利用EM算法构造了模型参数的最大扩展拟似然估计和最大伪似然估计.最后,通过随机模拟和实例研究,结果表明模型和方法的有效性和有用性.     

First hitting time models for the generalized inverse Gaussian distribution

Any generalized inverse Gaussian distribution with a non-positive power parameter is shown to be the distribution of the first hitting time of level 0 for each of a variety of time-homogeneous diffusions on the interval [0, ∞). The infinite divisibility of the generalized inverse Gaussian distributions is a simple consequence of this and an elementary convolution formula for these distributions.     

Estimation in generalized linear models for functional data via penalized likelihood

We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frameworks.     

Fiducial generalized p-values for testing zero-variance components in linear mixed-effects models

Linear mixed-effects models are widely used in analysis of longitudinal data. However, testing for zero-variance components of random effects has not been well-resolved in statistical literature, although some likelihood-based procedures have been proposed and studied. In this article, we propose a generalized p-value based method in coupling with fiducial inference to tackle this problem. The proposed method is also applied to test linearity of the nonparametric functions in additive models. We provide theoretical justifications and develop an implementation algorithm for the proposed method. We evaluate its finite-sample performance and compare it with that of the restricted likelihood ratio test via simulation experiments. We illustrate the proposed approach using an application from a nutritional study.     

Validation tests for the innovation distribution in INAR time series models

Goodness-of-fit tests are proposed for the innovation distribution in INAR models. The test statistics incorporate the joint probability generating function of the observations. Special emphasis is given to the INAR(1) model and particular instances of the procedures which involve innovations from the general family of Poisson stopped-sum distributions. A Monte Carlo power study of a bootstrap version of the test statistic is included as well as a real data example. Generalizations of the proposed methods are also discussed.