Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Numerical Comparisons for the Null Distribution of the Bagai Statistic

  • Ha, Hyung-Tae (Department of Applied Statistics, Gachon University)
  • Received : 2011.12.12
  • Accepted : 2012.02.23
  • Published : 2012.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Bagai et al. (1989) proposed a distribution-free test for stochastic ordering in the competing risk model, and recently Murakami (2009) utilized a standard saddlepoint approximation to provide tail probabilities for the Bagai statistic under finite sample sizes. In the present paper, we consider the Gaussian-polynomial approximation proposed in Ha and Provost (2007) and compare it to the saddlepoint approximation in terms of approximating the percentiles of the Bagai statistic. We make numerical comparisons of these approximations for moderate sample sizes as was done in Murakami (2009). From the numerical results, it was observed that the Gaussianpolynomial approximation provides comparable or greater accuracy in the tail probabilities than the saddlepoint approximation. Unlike saddlepoint approximation, the Gaussian-polynomial approximation provides a simple explicit representation of the approximated density function. We also discuss the details of computations.

Keywords

References

  1. Bagai, I., Deshpande, J. V. and Kochar, S. C. (1989). Distribution free tests for stochastic ordering in the competing risks model, Biometrika, 76, 775-781. https://doi.org/10.1093/biomet/76.4.775
  2. Bean, R., Froda, S. and Van Eeden, C. (2004). The normal, edgeworth, saddlepoint, uniform approx-imations to the Wilcoxon-Mann-Whitney null-distribution: A numerical comparison, Journal of Nonparametric Statistics, 16, 279-288. https://doi.org/10.1080/10485250310001622677
  3. Butler, R. W. (2007). Saddlepoint Approximations with Applications, Cambridge University Press.
  4. Daniels, H. E. (1954). Saddlepoint approximations in statistics, Annals of Mathematical Statistics, 25, 631-650. https://doi.org/10.1214/aoms/1177728652
  5. Daniels, H. E. (1987). Tail probability approximations, International Statistical Review, 55, 37-48. https://doi.org/10.2307/1403269
  6. Easton, G. S. and Ronchetti, E. (1986). General saddlepoint approximations with applications to L statistics, Journal of the American Statistical Association, 81, 420-430.
  7. Goutis, C. and Casella, G. (1999). Explaining the saddlepoint approximation, American Statistician, 53, 216-224.
  8. Ha, H-T. (2007). Moment-based density approximation algorithm for symmetric distributions, The Korean Communications in Statistics, 14, 583-592. https://doi.org/10.5351/CKSS.2007.14.3.583
  9. 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
  10. Huzurbazar, S. (1999). Practical saddlepoint approximations, American Statistician, 53, 225-232.
  11. Jensen, J. L. (1995). Saddlepoint Approximations, Oxford University Press.
  12. Kolassa, J. E. (2006). Series Approximation Methods in Statistics, Springer-Verlag.
  13. Lugannani, R. and Rice, S. O. (1980). Saddlepoint approximation for the distribution of the sum of independent random variables, Advances in Applied Probability, 12, 475-490. https://doi.org/10.2307/1426607
  14. Murakami, H. (2009). A saddlepoint approximation to a distribution-free test for stochastic ordering in the competing risks model, REVSTAT - Statistical Journal, 7, 189-201.
  15. Provost, S. B., Min, J. and Ha, H-T. (2009). Moment-based approximations of probability mass functions with applications involving order statistics, Communication in Statistics: Theory and Methods, 38, 1969-1981 https://doi.org/10.1080/03610920902835052
  16. Reid, N. (1988). Saddlepoint methods and statistical inference (with discussion), Statistical Science, 3, 213-238. https://doi.org/10.1214/ss/1177012906
  17. Wood, A. T. A., Booth, J. G. and Butler, R. W. (1993). Saddlepoint approximations to the CDF of some statistics with nonnormal limit distributions, Journal of the American Statistical Association, 88, 680-686. https://doi.org/10.1080/01621459.1993.10476322