基于似然函数的统计推断

本项目研究基于似然函数的统计推断，着重于统计假设检验问题。提出了新的似然推断方法，主要以两种方式推广似然比检验，一是以原假设和全参数空间上的似然函数之比作为差异变量来求取预测p-值，二是用原假设和全参数空间上的似然函数的加权积分之比作为检验统计量，并研究它的Wilks现象。这两种方法都与后验预测分布有关，相信后验预测分布将在统计推断中发挥更大的作用。新的似然推断方法主要用于复杂的和高维的统计模型。研究方法涉及频率和Bayes两个统计学派。基于似然推断的重要地位，在这个大数据时代，分析复杂数据和高维数据将是统计学的一个主要任务，所以，本项目具有重要的科学意义和应用价值。

In this project statistical inference based on likelihood functions is considered. It is mainly concerned about hypothesis testing. New statistical inference approaches are proposed. In two ways the classical likelihood inferences are generalized. One is the posterior projective p-value in which the ratio of the likelihood functions on null hypothesis and full parameter space is taken as a discrepancy variable. Another is the test statistic constructed by the ratio of two weighted integrals of likelihood functions on null hypothesis and full parameter space. Meanwhile Wilks phenomenon for the test statistic is investigated. Both of two generalizations are relative to the posterior predictive distribution which should be of a great role in statistical inference. New likelihood inference method will be employed to analyses of the statistical models with complex and high dimensional data. The method is not only frequentist, but also Bayesian. Because the likelihood inference plays a dominant role in statistical inference and the analysis of complex or high dimensional data is an urgent work for statistics, the project is important both in theory and practice at the age of big data.