Research in Statistics

在这个项目中，首席调查员(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：(1)关于非高斯线性混合模型的部分观测信息和推断；(2)纵向数据分析的迭代加权最小二乘法；(3)相依观测的全局极大值检验；(4)广义线性混合模型诊断学。线性混合模型在实际中被广泛用于相关观测值，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，在实践中，常态假设很可能会被违反。众所周知，基于正态分布的方法，如REML，即使正态分布失效，仍能产生相容的估计量。然而，这些估计量的标准误差的估计是复杂的，因为对于非高斯数据，高斯估计量的渐近协方差矩阵包含额外的未知参数。此外，项目(Ii)提出了一种迭代加权最小二乘法来计算线性模型中回归系数的有效估计，并研究了它的性质；项目(Iii)旨在扩展I.D.情形下提出的检验似然方程的根是否对应于似然函数的全局最大值的方法；项目(Iv)发展了广义线性混合模型的拟合优度检验和非形式模型检验方法。例如，在医学研究中，随着时间的推移，重复的测量通常是从同一个人身上收集的。可以合理地假设，同一个体的观察结果之间存在相关性。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况。对于线性混合模型，已经发展了参考方法，但大多是在观测值为正态(或高斯)的假设下进行的。然而，这一重大假设在实践中很可能会遭到违反。PI旨在开发一种强大的方法来推断非高斯线性混合模型。对于广义线性混合模型，PI建议开发模型检验方法，以填补这些模型应用方面的一个重要空白。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.