High Order Adjusted Block-wise Empirical Likelihood For Weakly Dependent Data.

The upper limit on the coverage probability of the empirical likelihood ratio confidence region severely hampers its application in statistical inferences. The root cause of this upper limit is the convex hull of the estimating functions that is used in the construction of the profile empirical likelihood. For i.i.d data, various methods have been proposed to solve this issue by modifying the convex hull, but it is not clear how well these methods perform when the data is no longer independent. In this paper, we consider weakly dependent multivariate data, and we combine the block-wise empirical likelihood with the adjusted empirical likelihood to tackle data dependency and the convex hull constraint simultaneously. We show that our method not only preserves the much celebrated asymptotic $\chi^2-$distribution, but also improves the coverage probability by removing the upper limit. Further, we show that our method is also Bartlett correctable, thus is able to achieve high order asymptotic coverage accuracy.