Advertisement

Small Sample Asymptotics for Higher-Order Spacings

  • Riccardo Gatto
  • S. Rao Jammalamadaka
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

In this chapter, we give conditional representations for families of statistics based on higher-order spacings and spaing frequencies. This allows us to compute accurate approximations to the distribution of such statistics, including tail probabilities and critical values. These results generalize those discussed in (1999) and are essential in using such statistics in various testing contexts.

Keywords and phrases

Goodnes-of-fit tests nonparametric tests rank tests m-step spacings m-step spacing frequencies two-sample tests Dirichlet gamma negative binomial distributions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Daniels, H. E. (1954). Saddlepoint approximations in statistics, The Annals of Mathematical Statistics, 25, 631–650.MathSciNetGoogle Scholar
  2. 2.
    Darling, D. A. (1953). On a class of problems related to the random division of an interval, The Annals of Mathematical Statistics, 24, 239–253.MathSciNetGoogle Scholar
  3. 3.
    Del Pino, G. E. (1979). On the asymptotic distributionof k-spacings with applications to goodness-of-fit tests, The Annals of Statistics, 7, 1058–1065.zbMATHMathSciNetGoogle Scholar
  4. 4.
    Dixon, W. J. (1940). A criterion for testing the hypothesis that two samples are from the same population, Annals of Mathematical Statistics, 11, 199–204.MathSciNetGoogle Scholar
  5. 5.
    Gatto, R. (2001). Symbolic computation for approximating the distributions of some families of one and two-sample nonparametric test statistics, Statistics and Computing, 11, 449–455.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gatto, R., and Jammalamadaka, S. R. (1999). A conditional saddlepoint approximation for testing problems, Journal of the American Statistical Association, 94, 533–541.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Holst, L., and Rao, J. S. (1980). Asymptotic theory for some families of two-sample nonparametric statistics, Sankhyà, Series A, 42, 19–52.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Holst, L., and Rao, J. S. (1981). Asymptotic spacings theory with applications to the two-sample problem, The Canadian Journal of Statistics, 9, 79–89.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lugannani, R., and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables, Advances in Applied Probability, 12, 475–490.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jammalamadaka, S. R., and Schweitzer, R. L. (1985). On tests for the two-sample problem based on higher order spacing-frequencies, In Statistical Theory and Data Analysis (Ed., K. Matusita), pp. 583–618, North-Holland, Amsterdam.Google Scholar
  11. 11.
    Pyke, R. (1965). Spacings, The Journal of the Royal Statistical Society, Series B, 27, 395–449.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Rao, J. S. (1976). Some tests based on arc-lengths for the circle, Sankhyà, Series B, 4, 329–338.Google Scholar
  13. 13.
    Rao, J. S., and Kuo, M. (1984). Asymptotic results on the Greenwood statistic and some of its generalizations, The Journal of the Royal Statistical Society, Series B, 46, 228–237.zbMATHMathSciNetGoogle Scholar
  14. 14.
    Rao, J. S., and Sethuraman, J. (1975). Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors, The Annals of Statistics, 3, 299–313.zbMATHMathSciNetGoogle Scholar
  15. 15.
    Sethuraman, J., and Rao, J. S. (1970). Pitmann efficiencies of tests based on spacings, In Nonparametric Techniques in Statistical Inference (Ed., M. L. Puri), Cambridge University Press, Cambridge.Google Scholar
  16. 16.
    Wilks, S. S. (1962). Mathematical Statistics, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  17. 17.
    Zhou, X., and Jammalamadaka, S. R. (1989). Bahadur efficiencies of spacings test for goodness of fit, Annals of the Institute of Statistical Mathematics, 41, 541–553.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • Riccardo Gatto
    • 1
    • 2
  • S. Rao Jammalamadaka
    • 1
    • 2
  1. 1.University of BernBernSwitzerland
  2. 2.University of CaliforniaSanta BarbaraUSA

Personalised recommendations