Abstract
We study nonparametric estimation of the distribution function (DF) of a continuous random variable based on a ranked set sampling design using the exponentially tilted (ET) empirical likelihood method. We propose ET estimators of the DF and use them to construct new resampling algorithms for unbalanced ranked set samples. We explore the properties of the proposed algorithms. For a hypothesis testing problem about the underlying population mean, we show that the bootstrap tests based on the ET estimators of the DF are asymptotically normal and exhibit a small bias of order O(n−1). We illustrate the methods and evaluate the finite sample performance of the algorithms under both perfect and imperfect ranking schemes using a real data set and several Monte Carlo simulation studies. We compare the performance of the test statistics based on the ET estimators with those based on the empirical likelihood estimators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahn, S., Lim, J., & Wang, X. (2014). The Student’st approximation to distributions of pivotal statistics from ranked set samples. Journal of the Korean Statistical Society, 43(4), 643–652.
Amiri, S., Jafari Jozani, M., & Modarres, R. (2014). Resampling unbalanced ranked set sampling with application in testing hypothesis about the population mean. Journal of Agricultural, Biological, and Environmental Statistics, 19, 1–17.
Baklizi, A. (2009). Empirical likelihood intervals for the population mean and quantiles based on balanced ranked set samples. Statistical Methods and Applications, 18(4), 483–505.
Chaudhuri, S., & Ghosh, M. (2011). Empirical likelihood for small area estimation. Biometrika, 98(2), 473–480.
Chen, Z., Bai, Z., & Sinha, B. K. (2004). Ranked set sampling: theory and applications. New York: Springer-Verlag.
Dell, T. R., & Clutter, J. L. (1972). Ranked set sampling theory with order statistics background. Biometrics 545–555.
DiCiccio, T. J., & Romano, J. P. (1990). Nonparametric confidence limits by resampling methods and least favourable families. International Statistical Review, 58, 59–76.
Efron, B. (1981). Nonparametric standard errors and confidence intervals (with discussion). The Canadian Journal of Statistics, 9, 139–172.
Efron, B., & Tibshirani, R. (1993). An introduction to bootstrap. New York: Chapman & Hall.
Feuerveger, A., Robinson, J., & Wong, A. (1999). On the relative accuracy of certain bootstrap procedures. The Canadian Journal of Statistics, 27, 225–236.
Frey, J., Ozturk, O., & Deshpande, J. V. (2007). Nonparametric tests for perfect judgment rankings. Journal of the American Statistical Association, 102, 708–717.
Frey, J., & Wang, L. (2013). Most powerful rank tests for perfect rankings. Computational Statistics & Data Analysis, 60, 157–168.
Hall, P. (1992). The bootstrap and edgeworth expansion. New York: Springer-Verlag.
Jafari Jozani, M., & Johnson, B. C. (2012). Randomized nomination sampling for finite populations. Journal of Statistical Planning and Inference, 142, 2103–2115.
Kim, J. K. (2010). Calibration estimation using exponential tilting in sample surveys. Survey Methodology, 36(2), 145–155.
Kostov, P. (2012). Empirical likelihood estimation of the spatial quantile regression. Journal of Geographical Systems, 15(1), 51–69.
Li, T., & Balakrishnan, N. (2008). Some simple nonparametric methods to test for perfect ranking in ranked set sampling. Journal of Statistical Planning and Inference, 138, 1325–1338.
Liu, T., Lin, N., & Zhang, B. (2009). Empirical likelihood for balanced ranked-set sampled data. Science in China Series A: Mathematics, 52(6), 1351–1364.
Modarres, R., Hui, T. P., & Zhang, G. (2006). Resampling methods for ranked set samples. Computational Statistics & Data Analysis, 51, 1039–1050.
Owen, A. B. (2001). Empirical likelihood. Chapman and Hall/CRC.
Ozturk, O., & Jafari Jozani, M. (2014). Inclusion probabilities in partially rank ordered set sampling. Computational Statistics & Data Analysis, 69, 122–132.
Schennach, S. M. (2005). Bayesian exponentially-tilted empirical likelihood. Biometrika, 92, 31–46.
Schennach, S. M. (2007). Point estimation with exponentially tilted empirical likelihood. Annals of Statistics, 35(2), 634–672.
Stokes, S. L., & Sager, T. W. (1988). Characterization of a ranked-set sample with application to estimating distribution functions. Journal of the American Statistical Association, 83(402), 374–381.
Vock, M., & Balakrishnan, N. (2011). A Jonckheere-Terpstra-type test for perfect ranking in balanced ranked set sampling. Journal of Statistical Planning and Inference, 141, 624–630.
Wolfe, D. A. (2012). Ranked set sampling: its relevance and impact on statistical inference. In International scholarly research notices.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amiri, S., Jafari Jozani, M. & Modarres, R. Exponentially tilted empirical distribution function for ranked set samples. J. Korean Stat. Soc. 45, 176–187 (2016). https://doi.org/10.1016/j.jkss.2015.09.004
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1016/j.jkss.2015.09.004