Profile empirical likelihood for parametric and semiparametric models

This paper introduces a profile empirical likelihood and a profile conditionally empirical likelihood to estimate the parameter of interest in the presence of nuisance parameters respectively for the parametric and semiparametric models. It is proven that these methods propose some efficient estimators of parameters of interest in the sense of least-favorable efficiency. Particularly, for the decomposable semiparametric models, an explicit representation for the estimator of parameter of interest is derived from the proposed nonparametric method. These new estimations are different from and more efficient than the existing estimations. Some examples and simulation studies are given to illustrate the theoretical results. The first author is supported by NNSF projects (10371059 and 10171051) of China. The second author is supported by a grant from The Research Grants Council of the Hong Kong Special Administrative Region, China (#HKU7060/04P). The third author is supported by the University Research Committee of the University of Hong Kong and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU7323/01M).     

A statistical approach to graduation by mathematical formula

Graduation by mathematical formula is recast as problem of statistical estimation. The method of maximum likelihood is used to determine the estimates of the parameters. Theory is developed to allow for estimation without resorting to the usual ‘exposure’ formulas. Both single and multiple decrement models are considered. Theoretical results are obtained for some specific mortality models. Numerical procedures to obtain the estimates are considered.     

The geometric structure of the expected/observed likelihood expansions

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.     

Empirical likelihood for partial linear models

In this paper the empirical likelihood method due to Owen (1988,Biometrika,75, 237–249) is applied to partial linear random models. A nonparametric version of Wilks' theorem is derived. The theorem is then used to construct confidence regions of the parameter vector in the partial linear models, which has correct asymptotic coverage. A simulation study is conducted to compare the empirical likelihood and normal approximation based method. Research supported by NNSF of China and a grant to the first author for his excellent Ph.D. dissertation work in China. Research supported by Hong Kong RGC CERG No. HKUST6162/97P.     

Estimating Frailty Models via Poisson Hierarchical Generalized Linear Models

Frailty models extend proportional hazards models to multivariate survival data. Hierarchical-likelihood provides a simple unified framework for various random effect models such as hierarchical generalized linear models, frailty models, and mixed linear models with censoring. Wereview the hierarchical-likelihood estimation methods for frailty models. Hierarchical-likelihood for frailty models can be expressed as that for Poisson hierarchical generalized linear models. Frailty models can thus be fitted using Poisson hierarchical generalized linear models. Properties of the new methodology are demonstrated by simulation. The new method reduces the bias of maximum likelihood and penalized likelihood estimates.     

Bayesian Synthetic Likelihood

Having the ability to work with complex models can be highly beneficial. However, complex models often have intractable likelihoods, so methods that involve evaluation of the likelihood function are infeasible. In these situations, the benefits of working with likelihood-free methods become apparent. Likelihood-free methods, such as parametric Bayesian indirect likelihood that uses the likelihood of an alternative parametric auxiliary model, have been explored throughout the literature as a viable alternative when the model of interest is complex. One of these methods is called the synthetic likelihood (SL), which uses a multivariate normal approximation of the distribution of a set of summary statistics. This article explores the accuracy and computational efficiency of the Bayesian version of the synthetic likelihood (BSL) approach in comparison to a competitor known as approximate Bayesian computation (ABC) and its sensitivity to its tuning parameters and assumptions. We relate BSL to pseudo-marginal methods and propose to use an alternative SL that uses an unbiased estimator of the SL, when the summary statistics have a multivariate normal distribution. Several applications of varying complexity are considered to illustrate the findings of this article. Supplemental materials are available online. Computer code for implementing the methods on all examples is available at https://github.com/cdrovandi/Bayesian-Synthetic-Likelihood.     

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.     

A random model approach for the LASSO

The least absolute selection and shrinkage operator (LASSO) is a method of estimation for linear models similar to ridge regression. It shrinks the effect estimates, potentially shrinking some to be identically zero. The amount of shrinkage is governed by a single parameter. Using a random model formulation of the LASSO, this parameter can be specified as the ratio of dispersion parameters. These parameters are estimated using an approximation to the marginal likelihood of the observed data. The observed score equations from the approximation are biased and hence are adjusted by subtracting an empirical estimate of the expected value. After estimation, the model effects can be tested (via simulation) as the distribution of the observed data given that all model effects are zero is known. Two related simulation studies are presented that show that dispersion parameter estimation results in effect estimates that are competitive with other estimation methods (including other LASSO methods).     

协变量随机缺失下线性模型的经验似然推断及其应用

考虑协变量带有缺失的线性模型,提出了加权的经验似然方法和借补的经验似然方法,证明了所提出的经验对数似然比渐近于χ~2分布,由此构造回归系数的置信域。模拟研究了所提出方法的有限样本性质,并进行了实例分析。     

To be or to become: how dynamic geometry changes discourse

In this article, we investigate the impact of the introduction of a dynamic geometry environment on mathematical thinking by identifying changes in discourse engendered by its introduction in a high school geometry class. Our focus is on the teacher, and we find significant differences between static and dynamic geometry in terms of the ways in which the teacher talks about geometric objects, makes use of visual artifacts and models geometric reasoning. Even though these changes have major implications for the geometry being studied, they are made only very implicitly in the classroom.     

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.     

Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model

Abstract

Nonlinear mixed-effects models have received a great deal of attention in the statistical literature in recent years because of the flexibility they offer in handling the unbalanced repeated-measures data that arise in different areas of investigation, such as pharmacokinetics and economics. Several different methods for estimating the parameters in nonlinear mixed-effects model have been proposed. We concentrate here on two of them—maximum likelihood and restricted maximum likelihood. A rather complex numerical issue for (restricted) maximum likelihood estimation in nonlinear mixed-effects models is the evaluation of the log-likelihood function of the data, because it involves the evaluation of a multiple integral that, in most cases, does not have a closed-form expression. We consider here four different approximations to the log-likelihood, comparing their computational and statistical properties. We conclude that the linear mixed-effects (LME) approximation suggested by Lindstrom and Bates, the Laplacian approximation, and Gaussian quadrature centered at the conditional modes of the random effects are quite accurate and computationally efficient. Gaussian quadrature centered at the expected value of the random effects is quite inaccurate for a smaller number of abscissas and computationally inefficient for a larger number of abscissas. Importance sampling is accurate, but quite inefficient computationally.     

Bootstrapping Log Likelihood and EIC, an Extension of AIC

Akaike (1973, 2nd International Symposium on Information Theory, 267-281,Akademiai Kiado, Budapest) proposed AIC as an estimate of the expected loglikelihood to evaluate the goodness of models fitted to a given set of data.The introduction of AIC has greatly widened the range of application ofstatistical methods. However, its limit lies in the point that it can beapplied only to the cases where the parameter estimation are performed bythe maximum likelihood method. The derivation of AIC is based on theassessment of the effect of data fluctuation through the asymptoticnormality of MLE. In this paper we propose a new information criterion EICwhich is constructed by employing the bootstrap method to simulate the datafluctuation. The new information criterion, EIC, is regarded as an extensionof AIC. The performance of EIC is demonstrated by some numerical examples.     

Solving the Likelihood Equations

Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the Jukes–Cantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined.     

On Mascheroni's La geometria del compasso at the beginning of the 19th century

Lorenzo Mascheroni's 1797 work La geometria del compasso, which develops a geometry based solely on compass constructions, is considered by the author as stepping back behind the “demarcation line” of Euclidean geometry. In this work Mascheroni emphasizes the practical aspects of this geometry over a theoretical approach. A century later, in 1899, David Hilbert and his student Michael Feldblum proposed a totally different approach – algebraic and axiomatic – concerning geometric constructions based on various instruments. Taking into account that, at the end of the 18th century, straightedge geometry was also developed, one may ask what happened to the image of instrument-based geometry during the 19th century? By focusing on Mascheroni's book and its reception, this article aims to examine the various views and conceptions of mathematicians with respect to this geometry.     

右删失数据下非线性回归模型的经验似然推断

考虑随机右删失数据下非线性回归模型,提出了模型中未知参数的调整的经验对数似然比统计量.在一定的条件下,证明了.所提出的的统计量具有渐近χ~2分布,由此结果构造了兴趣参数的置信域.通过模拟研究,对经典的经验似然、调整的经验似然和非线性最小二乘方法在有限样本下进行了比较,并对氯离子浓度试验数据进行了分析.     

Semi- and Nonparametric Modeling of Ordinal Data

Parametric models for categorical ordinal response variables, like the proportional odds model or the continuation ratio model, assume that the predictor is given by a linear form of covariates. In this article the parametric models are extended to include smooth components in a semiparametric or partially parametric fashion. Parts of the covariates are thereby modeled linearly while other covariates are modeled as unspecified but smooth functions. Estimation is based on a combination of local likelihood and profile likelihood and asymptotic properties of the estimates are derived. In a simulation study it is demonstrated that the profile likelihood approach is to be preferred over a backfitting procedure. Two data examples demonstrate the applicability of the models.     

Tensors and likelihood expansions in the presence of nuisance parameters

Stochastic expansions of likelihood quantities are usually derived through oridinary Taylor expansions, rearranging terms according to their asymptotic order. The most convenient form for such expansions involves the score function, the expected information, higher order log-likelihood derivatives and their expectations. Expansions of this form are called expected/observed. If the quantity expanded is invariant or, more generally, a tensor under reparameterisations, the entire contribution of a given asymptotic order to the expected/observed expansion will follow the same transformation law. When there are no nuisance parameters, explicit representations through appropriate tensors are available. In this paper, we analyse the geometric structure of expected/observed likelihood expansions when nuisance parameters are present. We outline the derivation of likelihood quantities which behave as tensors under interest-respectign reparameterisations. This allows us to write the usual stochastic expansions of profile likelihood quantities in an explicitly tensorial form.     

基于吴方法的几何定理证明的恒等式方法

多年来通常认为以吴方法为代表的几何定理机器证明的坐标法给出的证明不可读,或不是图灵意义下的类人解答.其实,只要对吴氏的算法做不多的改进,即将命题的结论多项式表示为其条件多项式的线性组合,就能获得不依赖于理论、算法和大量计算过程的恒等式明证.这样的恒等式可以转化为其他更简明且更有直观几何意义的点几何形式或向量及其他形式,从而获得多种证明方法.这也证明了点几何恒等式明证方法对等式型几何命题的普遍有效性.