, Volume 42, Issue 1, pp 421–439 | Cite as

On distances and goodness-of-fit tests for detecting multimodal distributions

  • Zinoviy Landsman
  • Meir Rom


Various distances between distributions and between densities are considered. The corresponding goodness-of-fit tests derived from them are examined for their abilities to detect multimodal alternatives. It is found that many well known techniques fail to detect such alternatives, while others do better in terms of their power results. These are mainly the tests derived from the variational metric which are based on spacings and gaps.

Key Words

Goodness-of-fit Multimodal Distributions Variational Metric EDF Spacings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Borovkov AA (1984) Mathematical statistics. Nauka Moscow (In Russian)Google Scholar
  2. [2]
    Cressie N (1976) On the logarithms of high-order spacings. Biometrika 63:343–355Google Scholar
  3. [3]
    Cressie N (1978) The minimum of higher order gaps. Austrl J of Statist 19:132–143Google Scholar
  4. [4]
    Cressie N (1979) An optimal statistic based on higher order gaps. Biometrika 66:619–627Google Scholar
  5. [5]
    D'Agostino RB, Stephens MA (1986) Goodness-of-fit techniques. Dekker New YorkGoogle Scholar
  6. [6]
    Del Pino GE (1979) On the asymptotic distribution ofk-spacings with applications to goodness-of-fit tests. Ann Statist 7:1058–1065Google Scholar
  7. [7]
    Greenwood M (1946) The statistical study of infectious disease. JRSS A 109:85–110Google Scholar
  8. [8]
    Hall P (1986) On powerful distributional tests based on sample spacings J Multivar Anal 19:201–224Google Scholar
  9. [9]
    Kolmogorov AN (1933) Sulla determinazione empirica di una legge di distibuziane. Giorna Ist Attuari 4:83–91Google Scholar
  10. [10]
    Prakasa, Rao BLS (1983) Nonparametric functional estimation. Academic Press New YorkGoogle Scholar
  11. [11]
    Pyke R (1965) Spacings. JRSS B 27:395–449Google Scholar
  12. [12]
    Rayner JCW, Best DJ (1989) Smooth tests of goodness-of-fit. Oxford Univ Press New YorkGoogle Scholar
  13. [13]
    Read TRC, Cressie N (1988) Goodness-of-fit sttistics for discrete multivariate data. Springer New YorkGoogle Scholar
  14. [14]
    Reiss RD (1989) Approximate distributions of order statistics with applications to non-parametric statistics. Springer New YorkGoogle Scholar
  15. [15]
    Sahler W (1968) A survey on distribution-free statistics based on distances between distribution functions. Metrika 13:149–169Google Scholar
  16. [16]
    Sherman B (1950) A random variable related to spacings of sample values. Ann Math Statist 21:339–361Google Scholar
  17. [17]
    Sherman B (1957) Percentage ofW n statistic. Ann Math Statist 28:259–261Google Scholar
  18. [18]
    Van Ryzin J (1973) A histogram method of density estimation. Comm Statist 2(6):493–506Google Scholar

Copyright information

© Physica-Verlag 1995

Authors and Affiliations

  • Zinoviy Landsman
    • 1
  • Meir Rom
    • 1
  1. 1.Department of StatisticsUniversity of HaifaHaifaIsrael

Personalised recommendations