Abstract
In this paper, we discuss empirical likelihood-based inferences for the Lorenz curve. The profile empirical likelihood ratio statistics for the Lorenz ordinate are defined under the simple random sampling and the stratified random sampling designs. It is shown that the limiting distributions of the profile empirical likelihood ratio statistics are scaled Chi-square distributions with one degree of freedom. We also derive the limiting processes of the associated empirical likelihood-based Lorenz processes. Hybrid bootstrap and empirical likelihood intervals for the Lorenz ordinate are proposed based on the newly developed empirical likelihood theory. Extensive simulation studies are conducted to compare the relative performances of various confidence intervals for Lorenz ordinates in terms of coverage probability and average interval length. The finite sample performances of the empirical likelihood-based confidence bands are also illustrated in simulation studies. Finally, a real example is used to illustrate the application of the recommended intervals.
Similar content being viewed by others
References
Atkinson A.B. (1970) On the measurement of inequality. Journal of Economic Theory 2: 244–263
Beach C.M., Davidson R. (1983) Distribution-free statistical inference with Lorenz curves and income shares. Review of Economic Studies 50: 723–735
Belinga-Hill, N. (2007). Empirical likelihood confidence intervals for generalized Lorenz curve. Master thesis at Georgia State University, 38, Atlanta, GA, USA.
Chang R.K.R., Halfon N. (1997) Graphical distribution of pediatricians in the United States: An analysis of the fifty states and Washington, DC. Pediatrics 100: 172–179
Chen J.H., Qin J. (1993) Empirical likelihood estimation for finite populations and the effective usage of auxiliary information. Biometrika 80: 107–116
Chen S.X., Qin J. (2003) Empirical likelihood-based confidence intervals data with possible zero observations. Statistics & Probability Letters 65: 29–37
Chen S.X., Leung H.Y., Qin J. (2003) Information recovery in a study with surrogate endpoints. Journal of the American Statistical Association 98: 1052–1062
Csörgö M., Csörgö S., Horvath L. (1986) An asymptotic theory for empirical reliability and concentration process (Vol. 33). Springer, New York
Csörgö M., Gastwirth J., Zitikis R. (1998) Asymptotic confidence bands for the Lorenz and bonferroni curves based on the empirical Lorenz curve. Journal of statistical planning and inference 74: 65–91
Doiron D.J, Barrett G.F. (1996) Inequality in male and female earnings: the roles of hours and earnings. Review of Economics and Statistics 78: 410–420
Francisco C., Fuller W. (1991) Quantile estimation with a complex survey design. The Annals of Statistics 19: 454–469
Gail M.H., Gastwirth J.L. (1978) A scale-free goodness-of-fit test for the exponential distribution based on the Lorenz curve. Journal of the American Statistical Association 73: 229–243
Gastwirth J.L. (1971) A general definition of Lorenz curve. Econometrica 39: 1037–1039
Goldie C.M. (1977) Convergence theorems for empirical Lorenz curves and their inverses. Advances in Applied Probability 9: 765–791
Hall P., La Scala B. (1990) Methodology and algorithms of empirical likelihood. International Statistical Review 58: 109–127
Hall P., Owen A.B. (1993) Empirical likelihood confidence bands in density estimation. Journal of Computational and Graphical Statistics 2: 273–289
Hallas J., Støvring H. (2006) Templates for analysis of individual-level prescription data. Basic and Clinical Pharmacology and Toxicology 98: 260–265
Hart P.E. (1971) Entropy and other measures of concentration. Journal of the Royal Statistical Society: Series A 134: 73–89
Hill, M. (1992). The Panel Study of Income Dynamics: a user’s guide. Newbury Park, California/London, England: Sage Publications.
Lorenz M.C. (1905) Methods of measuring the concentration of wealth. Journal of the American Statistical Association 9: 209–219
Owen A. (1988) Empirical likelihood ratio confidence intervals for single functional. Biometrika 75: 237–249
Owen A. (1990) Empirical likelihood ratio confidence regions. The Annals of Statistics 18: 90–120
Owen A.B. (2001) Empirical Likelihood. Chapman & Hall/CRC, Noca Raton
Pollard, D. (1990). Empirical processes: Theory and applications. NSF-CBMS Regional Conference Series in Probability and Statistics (Vol. 2). Hayward, CA: Institute of Mathematical Statistics.
Sen A. (1973) On Economic inequality. Norton, New York
Wu C. (2004) Some algorithmic aspects of the empirical likelihood method in survey sampling. Statistica Sinica 14: 1057–1067
Zheng B. (2002) Testing Lorenz curves with non-simple random samples. Econometrica 70: 1235–1243
Zhong B., Rao J.N.K. (2000) Empirical likelihood inference under stratified random sampling using auxiliary population information. Biometrika 87: 929–938
Zhou X.H., Qin G.S., Lin H.Z., Li G. (2006) Inferences in censored cost regression models with empirical likelihood. Statistica Sinica 16: 1213–1232
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Qin, G., Yang, B. & Belinga-Hall, N.E. Empirical likelihood-based inferences for the Lorenz curve. Ann Inst Stat Math 65, 1–21 (2013). https://doi.org/10.1007/s10463-012-0355-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-012-0355-z