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Reduce the computation in jackknife empirical likelihood for comparing two correlated Gini indices. (English) Zbl 1432.62066

Authors’ abstract: The Gini index has been widely used as a measure of income (or wealth) inequality in social sciences. To construct a confidence interval for the difference of two Gini indices from the paired samples, D. Wang and Y. Zhao [Can. J. Stat. 44, No. 1, 102–119 (2016; Zbl 1357.62082)] used a profile jackknife empirical likelihood. However, the computing cost with the profile empirical likelihood could be very expensive. In this paper, we propose an alternative approach of the jackknife empirical likelihood method to reduce the computational cost. We also investigate the adjusted jackknife empirical likelihood and the bootstrap-calibrated jackknife empirical likelihood to improve coverage accuracy for small samples. Simulations show that the proposed methods perform better than Wang and Zhao’s methods in terms of coverage accuracy and computational time. Real data applications demonstrate that the proposed methods work very well in practice.

MSC:

62G05 Nonparametric estimation
62G15 Nonparametric tolerance and confidence regions

Citations:

Zbl 1357.62082
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References:

[1] Abdul-Sathar, E. I.; Jeevanand, E. S.; Nair, K. R.M., Bayesian Estimation of Lorenz Curve, Gini Index and Variance of Logarithms in a Pareto Distribution, Statistica, LXV, 193-205 (2005) · Zbl 1188.62133
[2] Biewen, M., Bootstrap Inference for Inequality, Mobility, and Poverty Measurement, Journal of Econometrics, 108, 2, 317-342 (2002) · Zbl 1020.62118 · doi:10.1016/S0304-4076(01)00138-5
[3] Chen, J.; Variyath, A. M.; Abraham, B., Adjusted Empirical Likelihood and Its Properties, Journal of Computational and Graphical Statistics, 17, 426-443 (2008) · doi:10.1198/106186008X321068
[4] Chotikapanich, D.; Griffiths, W. E., Estimating Lorenz Curve Using a Dirichlet Distribution, Journal of Business and Economic Statistics, 20, 290-295 (2002) · doi:10.1198/073500102317352029
[5] Davidson, R., Reliable Inference for the Gini Index, Journal of Econometrics, 150, 30-40 (2009) · Zbl 1429.91243 · doi:10.1016/j.jeconom.2008.11.004
[6] Gastwirth, J. L., The Estimation of the Lorenz Curve and Gini Index, The Review of Economics and Statistics, 54, 306-316 (1972) · doi:10.2307/1937992
[7] Giles, D. E.A., Calculating a Standard Error for the Gini Coeficient: Some Further Results, Oxford Bulletin of Economics and Statistics, 66, 425-433 (2004) · doi:10.1111/j.1468-0084.2004.00086.x
[8] Hjort, N.; Mckeague, I.; Van Keilegom, I., Extending the Scope of Empirical Likelihood, The Annals of Statistics, 37, 3, 1079-1111 (2009) · Zbl 1160.62029 · doi:10.1214/07-AOS555
[9] Hoeffding, W., A Class of Statistics with Asymptotically Normal Distribution, Annals of Mathematical Statistics, 19, 293-325 (1948) · Zbl 0032.04101 · doi:10.1214/aoms/1177730196
[10] Jing, B. Y.; Yuan, J. Q.; Zhou, W., Jackknife Empirical Likelihood, Journal of the American Statistical Association, 104, 1224-1232 (2009) · Zbl 1388.62136 · doi:10.1198/jasa.2009.tm08260
[11] Karagiannis, E.; Kovacevic, M., A Method to Calculate the Jackknife Variance Estimator for the Gini Coefficient, Oxford Bulletin of Economics and Statistics, 62, 119-122 (2000) · doi:10.1111/1468-0084.00163
[12] Li, M.; Peng, L.; Qi, Y., Reduce Computation in Profile Empirical Likelihood Method, The Canadian Journal of Statistics, 39, 2, 370-384 (2011) · Zbl 1271.62072 · doi:10.1002/cjs.10101
[13] Li, Z.; Xu, J.; Zhou, W., On Non-smooth Estimating Functions Via Jackknife Empirical Likelihood, The Scandinavian Journal of Statistics, 43, 49-69 (2016) · Zbl 1371.62028 · doi:10.1111/sjos.12164
[14] Mills, J. A.; Zandvakili, S., Statistical Inference Via Bootstrapping for Measures of Inequality, Journal of Applied Econometrics, 12, 2, 133-150 (1997) · doi:10.1002/(SICI)1099-1255(199703)12:2<133::AID-JAE433>3.0.CO;2-H
[15] Modarres, R.; Gastwirth, J. L., A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality, Oxford Bulletin of Economics and Statistics, 68, 385-390 (2006) · doi:10.1111/j.1468-0084.2006.00167.x
[16] Moothatu, T. S.K., Distribution of Maximum Likelihood Estimators of Lorenz Curves and Gini Index of Exponential Distribution, Annals of the Institute of Statistical Mathematics, 37, 473-479 (1985) · Zbl 0584.62023 · doi:10.1007/BF02481115
[17] Moothatu, T. S.K., The Best Estimator and a Strongly Consistent Asymptotically Normal Unbiased Estimator of Lorenz Curves, Gini Index and Their Entropy Index for Pareto Distribution, Sankya, Ser. B, 52, 115-127 (1990) · Zbl 0717.62019
[18] Ogwang, T., A Convenient Method of Computing the Gini Index and Its Standard Error, Oxford Bulletin of Economics and Statistics, 62, 123-129 (2000) · doi:10.1111/1468-0084.00164
[19] Owen, A., Empirical Likelihood (2001), London: Chapman and Hall, London · Zbl 0989.62019
[20] Peng, L., Empirical Likelihood Methods for the Gini Index, Australian and New Zealand Journal of Statistics, 53, 131-139 (2011) · Zbl 1241.91101 · doi:10.1111/j.1467-842X.2011.00614.x
[21] Qin, Y.; Rao, J. N.K.; Wu, C., Empirical Likelihood Confidence Intervals for the Gini Measure of Income Inequality, Economic Modeling, 27, 1429-1435 (2010) · doi:10.1016/j.econmod.2010.07.015
[22] Sang, Y.; Dang, X.; Zhao, Y., Jackknife Empirical Likelihood Methods for Gini Correlations and Their Equality Testing, Journal of Statistical Planning and Inference, 199, 45-59 (2019) · Zbl 1418.62221 · doi:10.1016/j.jspi.2018.05.004
[23] Shi, X., The Approximate Independence of Jackknife Pseudo-values and the Bootstrap Methods, Journal of Wuhan Institute of Hydraulic and Electrical Engineering, 2, 83-90 (1984)
[24] Summers, R.; Heston, A., The Penn World Tables (1995), Cambridge, MA: National Bureau of Economic Research
[25] Wang, D.; Zhao, Y., Jackknife Empirical Likelihood for Comparing Two Gini Indices, The Canadian Journal of Statistics, 44, 1, 102-119 (2016) · Zbl 1357.62082 · doi:10.1002/cjs.11275
[26] Wang, D.; Zhao, Y.; Gilmore, D. W., Jackknife Empirical Likelihood Confidence Interval for the Gini Index, Statistics & Probability Letters, 110, 289-295 (2016) · Zbl 1383.62328 · doi:10.1016/j.spl.2015.09.026
[27] Yitzhaki, S., Calculating Jackknife Variance Estimators for Parameters of the Gini Method, Journal of Business & Economic Statistics, 9, 235-239 (1991)
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