Empirical Likelihood Confidence Intervals for Lorenz Curve with Length-Biased Data

The Lorenz curve (LC) is the most fundamental and remarkable tool for processing the size distribution of income and wealth. The LC method is applied as a means to describe distributional consideration in economic analysis. On the other hand, the importance of the biased sampling problem has been well-recognized in statistics and econometrics. In this paper, the empirical likelihood (EL) procedure is proposed to make inferences about the LC in the length-biased setting. The limiting distribution of the EL-based log-likelihood ratio leads to a scaled Chi-square. This limiting distribution will be utilized to construct the EL ratio confidence interval for the LC. Another EL-based confidence interval is proposed by using the influence function method. Simulation studies are conducted to compare the performances of these EL-based confidence intervals with their counterparts in terms of coverage probability and average length. Real data analysis has been used to illustrate the theoretical results.