Confidence Intervals for the Mean Based on Exponential Type Inequalities and Empirical Likelihood

For independent observations, recently, it has been proposed to construct the confidence intervals for the mean using exponential type inequalities. Although this method requires much weaker assumptions than those required by the classical methods, the resulting intervals are usually too large. Still in special cases, one can find some advantage of using bounded and unbounded Bernstein inequalities. In this paper, we discuss the applicability of this approach for dependent data. Moreover, we propose to use the empirical likelihood method both in the case of independent and dependent observations for inference regarding the mean. The advantage of empirical likelihood is its Bartlett correctability and a rather simple extension to the dependent case. Finally, we provide some simulation results comparing these methods with respect to their empirical coverage accuracy and average interval length. At the end, we apply the above described methods for the serial analysis of a gene expression (SAGE) data example. 1. Introduction Although the classical -test is based on the assumption that observations are normally distributed, it is well known that it is a robust test and works well at least for symmetric distributions (see [1]). However, for skewed or heavy tailed distributions, the confidence intervals for the mean based on the inversion of the -test may give poor coverage. Using exponential type inequalities (such as Bernstein’s inequality), Rosenblum and van der Laan [2] presented a simple approach for constructing confidence intervals for the population mean. Rosenblum and van der Laan [2] dealt with the bounded version of Bernstein’s inequality which is applicable only for a few distributions such as, the bounded uniform distribution. To use it for general distributions, we need to use the unbounded version of Bernstein’s inequality, which was analyzed by Shilane et al. [3] for the negative binomial distribution. The confidence intervals based on exponential type inequalities have a guaranteed coverage probability under much weaker assumptions than required by the standard methods. Although the obtained confidence intervals are usually too large, there are situations when they give better coverage accuracy than the classical methods. In this paper, our goal is to use the empirical likelihood method introduced by Owen [4, 5] for the inference on the mean and compare it with the Rosenblum and van der Laan [2] method. The empirical likelihood method is based on the nonparametric likelihood ratio statistic, which similarly to the parametric case has