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。

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.