  Traditional meta-analysis assumes that the effect sizes estimated in
individual studies follow a Gaussian distribution. However, this distributional
assumption is not always satisfied in practice, leading to potentially biased
results. In the situation when the number of studies, denoted as K, is large,
the cumulative Gaussian approximation errors from each study could make the
final estimation unreliable. In the situation when K is small, it is not
realistic to assume the random-effect follows Gaussian distribution. In this
paper, we present a novel empirical likelihood method for combining confidence
intervals under the meta-analysis framework. This method is free of the
Gaussian assumption in effect size estimates from individual studies and from
the random-effects. We establish the large-sample properties of the
non-parametric estimator, and introduce a criterion governing the relationship
between the number of studies, K, and the sample size of each study, n_i. Our
methodology supersedes conventional meta-analysis techniques in both
theoretical robustness and computational efficiency. We assess the performance
of our proposed methods using simulation studies, and apply our proposed
methods to two examples.
