Abstract
We discuss jackknife empirical likelihood (JEL) and adjusted jackknife empirical likelihood (AJEL) based inference for finding confidence intervals for probability weighted moment (PWM). We obtain the asymptotic distribution of the JEL ratio and AJEL ratio statistics. We compare the performance of the proposed confidence intervals with recently developed methods in terms of coverage probability and average width. We also develop JEL and AJEL based tests for PWM and study its properties. Finally we illustrate our method using rainfall data of Indian states.
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Chen, J., Variyath, A. M., & Abraham, B. (2008). Adjusted empirical likelihood and its properties. Journal of Computational and Graphical Statistics, 17, 426–443.
David, H. A., & Nagaraja, H. N. (2003). Order Statistics. New York: John Wiley and Sons.
Greenwood, J. A., Landwehr, J. M., Matalas, N. C., & Wallis, J. R. (1979). Probability weighted moments. Water Resource Research, 15, 1055–1064.
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. The annals of mathematical statistics, 19, 293–325.
Hosking, J. R. M., Wallis, J. R., & Wood, E. F. (1985). Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27, 251–261.
Jing, B. Y., Yuan, J., & Zhou, W. (2009). Jackknife empirical likelihood. Journal of the American Statistical Association, 104, 1224–1232.
Lee, A. J. (1990). U-Statistics: Theory and Practice. New York: Marcel Dekker Inc.
Lehmann, E. L. (1951). Consistency and unbiasedness of certain nonparametric tests. The Annals of Mathematical Statistics, 22, 165–179.
Owen, A. (1990). Empirical likelihood ratio confidence regions. The Annals of Statistics, 77, 90–120.
Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, 237–249.
Owen, A. B. (2001). Empirical Likelihood. New York: Chapman & Hall/CRC.
Peng, L. (2011). Empirical likelihood methods for the Gini index. Australian & New Zealand Journal of Statistics, 53, 131–139.
Qin, G., Yang, B., & Hall, N. E. B. (2013). Empirical likelihood based inferences for Lorenz curve. Annals of Institute of Statistical Mathematics, 65, 1–21.
Qin, Y., Rao, J. N. K., & Wu, C. (2010). Empirical likelihood confidence intervals for the Gini measure of income inequality. Economic Modelling, 27, 1429–1435.
Shi, J., & Lau, T. S. (2000). Empirical likelihood for partially linear models. Journal of Multivariate analysis, 72, 132–148.
Shi, X. (1984). The approximate independence of jackknife pseudo-values and the bootstrap methods. Journal of Wuhan Institute Hydra-Electric Engineering, 2, 83–90.
Thomas, D. R., & Grunkemeier, G. L. (1975). Confidence interval estimation of survival probabilities for censoblue data. Journal of the American Statistical Association, 70, 865–871.
Tsao, M. (2004). Bounds on coverage probabilities of the empirical likelihood ratio confidence regions. The Annals of Statistics, 32, 1215–1221.
Vexler, A., Zou, L., & Hutson, A. D. (2017). An extension to empirical likelihood for evaluating probability weighted moments. Journal of Statistical Planning and Inference, 182, 50–60.
Wang, D., & Zhao, Y. (2016). Jackknife empirical likelihood for comparing two Gini indices. Canadian Journal of Statistics, 44, 102–119.
Wang, D., Zhao, Y., & Gilmore, D. W. (2016). Jackknife empirical likelihood confidence interval for the Gini index. Statistics & Probability Letters, 110, 289–295.
Vexler, A., & Yu, J. (2018). Empirical Likelihood Methods in Biomedicine and Health. CRC Press, Boca Raton: Chapman and Hall.
Whang, Y. J. (2006). Smoothed empirical likelihood methods for quantile regression models. Econometric Theory, 22, 173–205.
Zhao, Y., Meng, X., & Yang, H. (2015). Jackknife empirical likelihood inference for the mean absolute deviation. Computational Statistics & Data Analysis, 91, 92–101.
Zhou, M. (2015). Empirical likelihood method in survival analysis. Boca Raton: Chapman and Hall, CRC Press.
Acknowledgements
We thank the anonymous referees for their constructive suggestions that help to improve the presentation of the paper substantially. Deepesh Bhati would like to thanks Indian Statistical Institute, Chennai for the support during his visit to ISI Chennai.
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Bhati, D., Kattumannil, S.K. & Sreelakshmi, N. Jackknife empirical likelihood based inference for probability weighted moments. J. Korean Stat. Soc. 50, 98–116 (2021). https://doi.org/10.1007/s42952-020-00062-9
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DOI: https://doi.org/10.1007/s42952-020-00062-9