Data containing many zeroes is popular in statistical applications, such as survey data. A confidence interval based on the traditional normal approximation may lead to poor coverage probabilities, especially when the nonzero values are highly skewed and the sample size is small or moderately large. The empirical likelihood (EL), a powerful nonparametric method, was proposed to construct confidence intervals under such a scenario. However, the traditional empirical likelihood experiences the issue of under-coverage problem which causes the coverage probability of the EL-based confidence intervals to be lower than the nominal level, especially in small sample sizes. In this paper, we investigate the numerical performance of three modified versions of the EL: the adjusted empirical likelihood, the transformed empirical likelihood, and the transformed adjusted empirical likelihood for data with various sample sizes and various proportions of zero values. Asymptotic distributions of the likelihood-type statistics have been established as the standard chi-square distribution. Simulations are conducted to compare coverage probabilities with other existing methods under different distributions. Real data has been given to illustrate the procedure of constructing confidence intervals.

包含许多零的数据在统计应用程序中很普遍，例如调查数据。基于传统正态近似的置信区间可能导致较差的覆盖概率，尤其是当非零值高度偏斜并且样本大小较小或中等较大时。经验似然法（EL）是一种强大的非参数方法，被提出来构造这种情况下的置信区间。但是，传统的经验可能性会遇到覆盖不足问题，这会导致基于EL的置信区间的覆盖概率低于名义水平，尤其是在小样本量中。在本文中，我们研究了EL的三个修改版本的数值性能：调整后的经验似然，转换后的经验似然，以及具有各种样本大小和各种比例的零值的数据的变换后调整后的经验似然。似然类型统计量的渐近分布已建立为标准卡方分布。进行模拟以比较覆盖率与其他现有方法在不同分布下的覆盖率。实际数据已被用来说明构建置信区间的过程。