Research in Statistics

在这个项目中，主要研究者（PI）提出了四个重要主题的研究，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）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法;（iii）相依观测的全局最大值检验;（iv）广义线性混合模型诊断。线性混合模型在相关观测中有着广泛的应用，其典型假设是观测值服从正态分布。然而，正常假设在实践中很可能被违反。众所周知，基于正态性的方法，如REML，即使正态性失败，仍然会产生一致性检验。然而，对于非高斯数据，由于高斯估计的渐近协方差矩阵包含了额外的未知参数，因此这些估计的标准误差的估计是复杂的，Project（i）旨在彻底解决这一长期存在的实际问题。此外，项目（ii）提出了一种计算纵向数据分析线性模型中回归系数有效估计的迭代加权最小二乘方法，并研究了其性质.项目（iii）旨在推广www.example.com上提出的检验似i.i.d.case然方程的根是否与似然函数的全局最大值相对应的方法.项目（iv）发展了一种检验似然方程的根是否与似然函数的全局最大值相对应的方法.广义线性混合模型的拟合检验和非形式化模型检验方法。例如，在医学研究中，随着时间的推移，经常从同一个人那里收集重复的测量结果。假设来自同一个体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的情形。对于线性混合模型，已经开发了参考方法，但大多数是在假设观测值是正态（或高斯）的情况下。然而，在实践中，这一假设很可能被违背。PI的目的是发展一个强大的方法推断非高斯线性混合模型。对于广义线性混合模型，PI建议开发模型检查方法，填补了这些模型应用中的一个重要空白。开发的方法可能会对其他统计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.