| Empirical likelihood (EL) is a nonparametric likelihood approach for parameter estimation and hypothesis testing. A desirable feature of the EL method is that it allows Bartlett correction, which is a simple statistical adjustment on the test statistic to construct confidence regions with improved coverage accuracies. Previous studies have demonstrated the Bartlett correctability of EL for independent and identically distributed data and Gaussian short-memory time series. However, it is still unknown whether EL is Bartlett correctable for long-memory or non-Gaussian distributed time series. In this thesis, we establish the validity of the Edgeworth expansion for the signed root empirical log-likelihood ratio statistic to prove that EL is Bartlett correctable for Gaussian long-memory and non-Gaussian short-memory time series. For Gaussian long-memory time series, the Edgeworth expansion admits an irregular form with a power series of order log3 n/ [square root of n]. Based on the expansion, the coverage error of the EL confidence region can be reduced from O(log 6 n/n) to O(log3 n/n). For non-Gaussian short-memory time series, by carefully calculating the higher-order cumulants of the signed root empirical log-likelihood ratio statistic, the valid Edgeworth expansion can be established as a power series of order O(n-1/2). Based on the expansion, the coverage error of the EL confidence region can be reduced from O(n-1) to O (n-2) using the Bartlett correction technique. |
