Statistical inference with applications

近年来，基于似然的高阶渐近方法得到了广泛的研究，以准确地近似检验感兴趣的标量参数的p值。这些方法的一个问题是获得给定参数值的约束最大似然估计。在某些情况下，标准软件可能会报告一个最优值，但实际上并不是真正的最优值，或者方法完全无法收敛。Goffe et al.（1994）表明，模拟退火算法可以发现传统软件遗漏的最优。第一个项目的目的是将模拟退火技术应用到R中，以获得无约束和有约束的最大似然估计。因此，对于任何给定的参数模型，感兴趣的标量参数的p值都可以精确地近似。所提出的方法可以应用于可靠性、生存数据分析、时间序列分析和计量经济学中，其中感兴趣的参数可能没有封闭形式。当前基于似然的高阶渐近方法的一个局限性是它们仅适用于感兴趣的标量参数。然而，在许多常见的统计问题中，感兴趣的参数是一个向量。这些问题的常用方法通常只有一阶收敛性。第二个项目的目的是将高阶方法与方向检验相结合，以获得一个近似的p值，用于在一般模型设置中测试感兴趣的向量参数。所提出的方法的理论精度将被确定。应用包括一般的Behrens-Fisher问题，检验一般混合模型的方差齐性，以及结构方程模型。前两个项目依赖于全似然函数的存在。当全似然函数太复杂而无法处理时，是否可以使用复合似然函数？众所周知，在这种模型错误规范的形式下，对数复合似然比统计量的渐近分布涉及加权独立卡方变量的线性组合。第三个项目的目的是通过鞍点近似获得加权独立卡方变量的渐近分布。该结果使我们能够研究用于变化点检测的加权双部分和统计量的渐近分布。最后一个项目偏离了前三个提议的项目。其目的是应用贝叶斯方法来计算文档与用户任务相关的几率，以便检索更加个性化和准确的搜索结果。所提出的研究将产生新的大数据信息检索技术和工具。这些工具将导致更有效的信息检索应用，将给社会带来广泛的利益。

In recent year, likelihood-based higher order asymptotic methods have been extensively studied to accurately approximate the p-value for testing a scalar parameter of interest. One problem of these methods is to obtain the constrained maximum likelihood estimate for a given value of the parameter of interest. In some cases, standard software may either report an optimal value, but in fact, is not the actual optimum, or the methods completely failed to converge. Goffe et al. (1994) showed that the simulated annealing algorithm could uncover optima missed by traditional software. The aim of the first project is to implement the simulated annealing technique into R to obtain both the unconstrained and the constrained maximum likelihood estimates. Hence, the p-value of a scalar parameter of interest for any given parametric model can be accurately approximated. The proposed method can be applied to reliability, survival data analysis, time series analysis, and econometrics where the parameter of interest may not have a closed form.A limitation of the current likelihood-based higher order asymptotic methods is that they are only applicable to a scalar parameter of interest. However, in many common statistics problems, the parameter of interest is a vector. The common approaches for these problems usually have only first order convergence. The aim of the second project is to combine the higher order method with the direction test to obtain an approximate p-value for testing a vector parameter of interest in a general model setting. The theoretical accuracy of the proposed method will be determined. Applications include the general Behrens-Fisher problem, testing for homogeneity of variance for general mixed model, and structural equation models.The first two projects depend on the existence of the full likelihood function. When the full likelihood function is too complex to deal with, can the composite likelihood function be used? It is well-known that under this form of model mis-specification, the asymptotic distribution of the log composite likelihood ratio statistic involves a linear combination of weighted independent chi-square variates. The aim of the third project is to obtain the asymptotic distribution of the weighted independent chi-square variates via the saddlepoint approximation. The result allow us to study the asymptotic distribution of the weighted double partial sum statistic for change point detection.The last project deviates from the first three proposed projects. The aim is to apply the Bayesian approach to calculate the odds of a document being relevant with respect to the user task such that more personalized and accurate search results can be retrieved. The proposed research will generate novel information retrieval techniques and tools over big data. These tools will lead to more effective information retrieval applications which will bring broad benefits to the society.