Research in Statistics

在这个项目中，主要研究者（PI）提出了四个重要课题的研究，旨在开发相关观测情况下的统计推断和模型诊断方法。这些主题是：(i)关于非高斯线性混合模型的部分观测信息和推断；（ii）纵向数据分析的迭代加权最小二乘程序；依赖观测值的全球最大值检验；广义线性混合模型诊断。线性混合模型在相关观测中得到广泛应用。关于这些模型的一个典型假设是观测值是正态分布的。然而，常态假设在实践中很可能被违背。众所周知，即使正态性失败，基于正态性的方法（如REML）仍然会产生一致性估计器。然而，这些估计量的标准误差估计是复杂的，因为对于非高斯数据，高斯估计量的渐近协方差矩阵涉及额外的未知参数。项目(i)旨在彻底解决这一长期存在的现实问题。此外，项目（ii）提出了一个迭代加权最小二乘程序，用于计算纵向数据分析线性模型中回归系数的有效估计，并研究了其性质。项目（iii）旨在扩展在i.i.d案例中开发的一种方法，用于检查似然方程的根是否对应于似然函数的全局最大值。项目（iv）开发广义线性混合模型的拟合优度检验和非正式模型检验方法。相关响应在实践中经常遇到。例如，在医学研究中，经常从同一个人收集重复的测量数据。假设同一个人的观察结果之间存在相关性是合理的。线性和广义线性混合模型是两类重要的统计模型，广泛应用于相关观测。对于线性混合模型的推理方法已经开发，但大多是在假设观测值是正态（或高斯）的情况下。然而，在实践中，极大假设很可能被违背。该项目旨在开发一种强大的非高斯线性混合模型的推理方法。对于广义线性混合模型，PI建议开发模型检查方法，以填补这些模型应用中的重要空白。所开发的方法可能会对统计的其他领域以及生物医学研究、遗传学、生物学、经济学、教育和社会科学等领域产生影响。

In this project, the principle investigator (PI) proposesresearch on four important topics aimed to develop methodsof statistical inference and model diagnostics in cases ofcorrelated observations. These topics are: (i) partiallyobserved information and inference about non-Gaussianlinear mixed models; (ii) iterative weighted least squaresprocedures for analysis of longitudinal data; (iii) A testof global maximum for dependent observations; and (iv)generalized linear mixed model diagnostics. Linear mixedmodels are widely used in practice for correlated observations.A typical assumption regarding these models is that theobservations are normally distributed. However, the normalityassumption is likely to be violated in practice. It is known thatnormality based methods such as REML still produce consistentestimators even if normality fails. However, the estimationof standard errors of these estimators is complicated, becausefor non-Gaussian data the asymptotic covariance matrix of theGaussian estimator involves additional unknown parameters.Project (i) aims to completely solve this long-standing problemof practical interest. Furthermore, project (ii) proposes aniterative weighted least squares procedure for computingefficient estimators of the regression coefficients in linearmodels for longitudinal data analysis and studies its properties.Project (iii) aims to extend a method developed in the i.i.d.case for checking whether a root to the likelihood equationcorresponds to the global maximum of the likelihood function.Project (iv) develops goodness-of-fit tests and methods ofinformal model checking for generalized linear mixed models.Correlated responses are often encountered in practice. Forexample, in medical studies repeated measures are often collectedfrom the same individuals over time. It would be reasonable toassume that correlations exist among the observations from thesame individual. Linear and generalized linear mixed models aretwo important classes of statistical models widely used in casesof correlated observations. For linear mixed models methods ofinference have been developed, but mostly under the assumptionthat the observations are normal (or Gaussian). However, thenormality assumption is likely to be violated in practice. ThePI aims to develop a powerful method for inference aboutnon-Gaussian linear mixed models. For generalized linear mixedmodels, the PI proposes to develop methods of model checkingthat fills an important gap in the applications of these models.The methods developed are likely to have impact in other fieldsof statistics as well as in fields such as biomedical research,genetics, biology, economics, education and social science.