Applications Of Empirical Likelihood In High Dimensional Statistical Inferences And Measurements Of Income Inequality

Empirical likelihood(EL)method is a very effective non-parametric statistical method,which was proposed by Owen in 1991,and this method has been widely used in various fields.With non-linear constraints,the calculation of the EL method are greatly increased.Therefore,Jing,Yuan and Zhou(2009)proposed the jackknife empirical likelihood(JEL)method,which inherited the advantages of the EL method and reduced the amount of calculation.In addition,in order to extend the EL method to high dimensional problems,Tang and Leng(2010)combined the penalty method and the EL method to construct the penalized empirical likelihood(PEL)method,and gave the asymptotic distribution of the PEL statistic and the asymptotic confidence interval.In this thesis,based on the above methods,we mainly consider the high dimensional parameter estimation in nonsmooth U-statistic structured estimating equations,the post-selection inference in high dimensional problems,and the inference of the income inequality index.Firstly,the U-statistic is a very important class of statistics,many common statistics and estimators are U-statistics,see for example,the quantile regression,the Wilcoxon rank regression,the linear transformation model,and so on.In this thesis,based on the JEL method,we conduct the statistical inference on high dimensional parameters in the estimating equation with nonsmooth U-statistic structure,and give the asymptotic distribution of the JEL statistic and asymptotic confidence intervals of parameters to be estimated.Secondly,in high dimensional problems,we usually use penalty methods to prevent model overfitting.After screening important feature variables,for testing the importance of the selected features,we need to do significance tests of their coefficients.However,a key issue is how to consider the relationship between the selection process and the inference process.Motivated by Lee et al.(2016),after the PEL selection procedure,we conduct a new framework to make the more accurate inferences for significance tests of selected parameters.Especially,through the Karush-Kuhn-Tucker condition of the PEL method,we can get its polygonal lemma,which is equivalent to the selection process of the PEL method.From this lemma,conditional on the selection event,the PEL statistic is bounded,that is,it asymptotically obeys the square of the conditional truncated normal distribution.Furthermore,the truncated distribution can be transformed into a uniform distribution with better properties for constructing asymptotic confidence intervals of selected variables.Finally,the income inequality is a very valuable issue in both practical applications and scientific research.Therefore,in order to compare the gap between different incomes more intuitively and effectively,many researchers have proposed a variety of indices to measure the income inequality,for example,Gini index,Bonferroni index and De Vergottini index,etc.Since these indices can all be written as special cases of a more general form,we can use the JEL method to estimate the difference of the indices with a unified form,and give its asymptotic distribution and confidence interval.