A bootstrap test for symmetry based on quantiles

Abstract

In this paper, we construct a nonparametric test for the symmetry assumption of an underling distribution based on the sample quantiles. Bootstrap re-sampling from a symmetric empirical distribution function is used to obtain the p-value of the test. The power of the new test statistic is compared with some well-known symmetry tests using a simulation study. The results show that the proposed test preserves its level and it has reasonable power properties on the family of distribution evaluated.

Keywords

• Symmetry,Quantile,Test Power,Bootstrap,Simulation

References

1. Cabilio, P. and Masaro, J. A simple test of symmetry about an unknown median, Canadian Journal of Statistics 24, 349-361, 1996.
2. Drikvandi, R., Modarres, R. and Jalilian, A. H. A bootstrap test for symmetry based on ranked set samples, Computational Statistics and Data Analysis 55, 1807-1814, 2011.
3. Gupta, M. K. An asymptotically nonparametric test of symmetry, The Annals of Mathe- matical Statistics 38(3), 849-866, 1967.
4. Huber, P. J. Robust statistics: A review, The Annals of Mathematical Statistics 43, 1041 1067, 1972.
5. Meintanis, S.G. and Ngatchou-Wandji, J. Recent tests for symmetry with multivariate and structured data, in: Jiang, J., Roussas, G. G. and Samaniego, F. J. (Eds.), Non-Parametric Statistical Methods and Related Topics: A Festschrift in honor of Professor P.K. Bhat- tacharya on the occasion of his 80th birthday (World Scientic, New Jersey, 2012,) 35-73.
6. Miao, W., Gel, Y. R., and Gastwirth, J. L. A New Test of Symmetry about an Unknown Median. Random Walk, Sequential Analysis and Related Topics - A Festschrift in Honor of Yuan-Shih Chow. Eds.: Agnes Hsiung, Cun-Hui Zhang, and Zhiliang Ying, (World Scien- tic Publisher, 2006).
7. Mira, A. Distribution-free test for symmetry based on Bonferroni's measure, Journal of Applied Statistics 26, 959-972, 1999.
8. Modarres, R. Efficient nonparametric estimation of a distribution function, Computational Statistics and Data Analysis 39, 75-95, 2002.

9. Munzel, U. Nonparametric methods for paired samples, Statistica Neerlandica 53, 277-286, 1999.
10. Ngatchou- Wandji, J. On testing for the nullity of some skewness coefficients, International Statatistics Review 74, 47-65, 2006.
11. Sering, R. J. Approximation Theorems of Mathematical Statistics (John Wiley and Sons Inc, 1980).
12. Staudte, R. G. Inference for quantile measures of skewness, TEST 23, 751-768, 2014.
13. Taylor, H. M. and Karlin, S. An Introduction to Stochastic Modeling (Academic Press, 1998).
14. Van der vaart, A. W. Asymptotic statistics (Cambridge University Press, 1998).
15. Zheng, T. and Gastwirth, J.L. On Bootstrap Tests of Symmetry About an Unknown Median, Journal of Data Science 8(3), 413-427, 2010.