Adjusted empirical likelihood for probability density functions under strong mixing samples

Abstract

To obtain a profile empirical likelihood ratio statistic is essentially to address a constrained maximization problem. However, when sample size is small, the optimal function may have no numerical solution, as the convex hull of the sample points may not contain the zero vector 0 as its interior point. The adjusted empirical likelihood (AEL) method introduced by Chen, Variyath, and Abraham (2008) is very useful to solve this problem. The innovation of this article is that we extend the AEL method to probability density functions (p.d.f.) under dependent samples, as there are few literatures using this method under dependent samples. It is shown that the AEL ratio statistic for a p.d.f. is asymptotically -type distributed under a mixing sample, which is used to obtain an AEL-based confidence interval for the p.d.f. Our simulations show that the AEL is faster to compute than unadjusted empirical likelihood and the coverage probability based on the AEL method is superior to the other two usual methods, namely the unadjusted empirical likelihood and the normal approximation methods, particularly under small sample sizes.