The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Polynomially Adjusted Normal Approximation to the Null Distribution of Ansari-Bradley Statistic

  • Ha, Hyung-Tae (Department of Applied Statistics, Kyungwon University) ;
  • Yang, Wan-Youn (Department of Applied Statistics, Kyungwon University)
  • Received : 20110400
  • Accepted : 20111100
  • Published : 2011.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

The approximation for the distribution functions of nonparametric test statistics is a significant step in statistical inference. A rank sum test for dispersions proposed by Ansari and Bradley (1960), which is widely used to distinguish the variation between two populations, has been considered as one of the most popular nonparametric statistics. In this paper, the statistical tables for the distribution of the nonparametric Ansari-Bradley statistic is produced by use of polynomially adjusted normal approximation as a semi parametric density approximation technique. Polynomial adjustment can significantly improve approximation precision from normal approximation. The normal-polynomial density approximation for Ansari-Bradley statistic under finite sample sizes is utilized to provide the statistical table for various combination of its sample sizes. In order to find the optimal degree of polynomial adjustment of the proposed technique, the sum of squared probability mass function(PMF) difference between the exact distribution and its approximant is measured. It was observed that the approximation utilizing only two more moments of Ansari-Bradley statistic (in addition to the first two moments for normal approximation provide) more accurate approximations for various combinations of parameters. For instance, four degree polynomially adjusted normal approximant is about 117 times more accurate than normal approximation with respect to the sum of the squared PMF difference.

Keywords

References

  1. Ansari, A. R. and Bradley, R. A. (1960). Rank-sum tests for dispersions, The Annals of Mathematical Statistics, 31, 1174-1189. https://doi.org/10.1214/aoms/1177705688
  2. Barton, D. E. and David, F. N. (1958). A test for birth order effects, Annals of Human Genetics, 22, 250-257. https://doi.org/10.1111/j.1469-1809.1958.tb01420.x
  3. Bean, R., Froda, S. and van Eeden, C. (2004). The normal, edgeworth, saddlepoint, uniform approximations to the Wilcoxon-Mann-Whitney null-distribution: A numerical comparison, Journal of Nonparametric Statistics, 16, 279-288. https://doi.org/10.1080/10485250310001622677
  4. Froda, S. and Eeden, C. V. (2000). A uniform saddlepoint expansion for the null-distribution of the Wilcoxon-Mann-Whitney statistic, The Canadian Journal of Statistics, 28, 137-149. https://doi.org/10.2307/3315496
  5. Ha, H-T. and Provost, S. B. (2007). A viable alternative to resorting to statistical tables, Communication in Statistics: Simulation and Computation, 36, 1135-1151. https://doi.org/10.1080/03610910701569671
  6. Lehmann, E. L. (1951). Consistency and unbiasedness of certain nonparametric tests, Annals of Mathematical Statistics, 22, 165-179. https://doi.org/10.1214/aoms/1177729639
  7. Mood, A. M. (1954). On the asymptotic efficiency of certain nonparametric two-sample tests, Annals of Mathematical Statistics, 25, 514-522. https://doi.org/10.1214/aoms/1177728719
  8. Provost, S. B., Jing, M. and Ha, H-T. (2009). Moment-based approximations of probability mass functions with applications involving order statistics, Communication in Statistics: Theory and Method, 38, 1969-1981. https://doi.org/10.1080/03610920902835052
  9. Terry, M. E. (1952). Some rank order tests which are most powerful against specific parametric alternatives, Annals of Mathematical Statistics, 23, 346-366. https://doi.org/10.1214/aoms/1177729381