Studies On The Bartlett Correction And Adjusted Empirical Likelihood

Large sample or asymptotic methods are common methods used in statistical inference, since the exact distributions of the estimators or test statistics are usually difficult to find, and we have to use their asymptotic distributions for statistical inference.The accuracy of the large sample methods based on the first order asymptotic distribution may be low. Therefore,one important problem in statistics is how to improve these methods.In this thesis,we consider two of the methods suggested to improve them:Bartlett correction and adjusted empirical likelihood.We first consider the Bartlett type correction to Wald test for the regression coefficient in Cox proportional hazard model.It's proved that the distribution of the Bartlett corrected test statistic converges to chi-square distribution in an order of o(n^-1).Monte-Carlo simulations are performed to verify the improvement of Bartlett corrected Wald test when the sample size is small.We then apply the adjusted empirical likelihood method to construct confidence regions for the regression coefficient in Cox proportional hazard model and show that the distribution of adjusted empirical likelihood statistic converges to chi-square distribution. Monte-Carlo simulations are also performed to verify the improvement of adjusted empirical likelihood confidence regions to that based on first order asymptotic distribution. Finally,we apply the idea of adjusted empirical likelihood method to other types of nonparametric likelihood.It's shown that the adjusted exponential empirical likelihood and adjusted exponential tilting statistics we suggested converge to chi-square distribution in an order of O(n^-2).Simulation studies we performed also show that the coverage probability of the confidence regions based on adjusted methods are more accurate than those based on unadjusted methods.