A New Nonparametric Test of Symmetry

  • Kaushik Ghosh


We present a new nonparametric test for symmetry about an unknown location and investigate its large sample properties. Asymptotic normality of the test statistic is established and an estimator of the asymptotic variance is also presented. Results of a simulation study and data analysis are also presented.


Stat Assoc Asymptotic Variance Math Stat Unknown Location Stat Plan Inference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahmad IA (1996) Modification of some goodness of fit statistics II: Two-sample and symmetry testing. Sankhyá: Indian J Stat Ser A 58(3):464–472MATHGoogle Scholar
  2. 2.
    Ahmad IA, Li Q (1997) Testing symmetry of an unknown density function by kernel method. J Nonparametr Stat 7(3):279–293CrossRefMathSciNetGoogle Scholar
  3. 3.
    Aki S (1987) On nonparametric tests for symmetry. Ann Inst Stat Math 39:457–472MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Andersen PK, Borgan O, Gill RD, Keiding N (1993) Statistical methods based on counting processes. Springer, New YorkGoogle Scholar
  5. 5.
    Antille A, Kersting G (1977) Tests for symmetry. Z Wahrsch Verw Gebiete 39:235–255MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Antille A, Kersting G, Zucchini W (1982) Testing symmetry. J Am Stat Assoc 77(379): 639–646MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Arcones MA, Gine E (1991) Some bootstrap tests for symmetry for univariate continuous distributions. Ann Stat 19(3):1496–1511MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Baklizi A (2003) A conditional distribution free runs test for symmetry. J Nonparametr Stat 15(6):713–718MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Baringhaus L, Henze N (1992) A characterization of new and consistent tests for symmetry. Commun Stat Theory Methods 21(6):1555–1566MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bhattacharya PK, Gastwirth JL, Wright AL (1982) Two modified Wilcoxon tests for symmetry about an unknown location parameter. Biometrika 69(2):377–382MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bjerkedal T (1960) Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am J Hyg 72:130–148Google Scholar
  12. 12.
    Bonett DG, Seier E (2003) Confidence intervals for mean absolute deviations. Am Statistcian 57(4):233–236MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Boos DD (1982) A test for asymmetry associated with the Hodges–Lehmann estimator. J Am Stat Assoc 77(379):647–651MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Butler CC (1969) A test for symmetry using the sample distribution function. Ann Math Stat 40(6):2209–2210MATHCrossRefGoogle Scholar
  15. 15.
    Cabana A, Cabana EM (2000) Tests of symmetry based on transformed empirical processes. Can J Stat 28(4):829–839MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Cabilio P, Masaro J (1996) A simple test of symmetry about an unknown median. Can J Stat 24(3):349–361MATHCrossRefGoogle Scholar
  17. 17.
    Cohen JP, Menjoge SS (1988) One-sample run tests of symmetry. J Stat Plan Inference 18:93–100MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    CsörgHo S, Heathcote CR (1987) Testing for symmetry. Biometrika 74(1):177–184MathSciNetGoogle Scholar
  19. 19.
    Davis CE, Quade D (1978) U-statistics for skewness and symmetry. Commun Stat Theory Methods 7(5):413–418CrossRefGoogle Scholar
  20. 20.
    Doksum KA (1975) Measures of location and asymmetry. Scand J Stat 2(1):11–22MATHMathSciNetGoogle Scholar
  21. 21.
    Doksum KA, Fenstad G, Aaberge R (1977) Plots and tests for symmetry. Biometrika 64(3):473–487MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ekström M, Jammalamadaka SR (2007) An asymptotically distribution-free test of symmetry. J Stat Plan Inference 137:799–810MATHCrossRefGoogle Scholar
  23. 23.
    Eubank RL, Iariccia VN, Rosenstein RB (1992) Testing symmetry about and unknown median, via linear rank procedures. J Nonparametr Stat 1(4):301–311MATHCrossRefGoogle Scholar
  24. 24.
    Fernholz LT (1983) Von Mises calculus for statistical functionals. Springer, New YorkMATHGoogle Scholar
  25. 25.
    Finch SJ (1977) Robust univariate test of symmetry. J Am Stat Assoc 72(358):387–392MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Gastwirth JL (1971) On the sign test for symmetry. J Am Stat Assoc 66(336):821–823MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ghosh K, Tiwari RC (2009) A unified approach to variations of ranked set sampling with applications. J Nonparametr Stat 21(4):471–485MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Gupta MK (1967) An asymptotically nonparametric test of symmetry. Ann Math Stat 38(3):849–866MATHCrossRefGoogle Scholar
  29. 29.
    Hájek J (1962) Asymptotically most powerful rank-order tests. Ann Math Stat 33:1124–1147MATHCrossRefGoogle Scholar
  30. 30.
    Hájek J, Sidák Z (1967) Theory of rank tests. Academic, New YorkMATHGoogle Scholar
  31. 31.
    Henze N (1993) On the consistency of a test for symmetry based on a runs statistic. J Nonparametr Stat 3(2):195–199CrossRefMathSciNetGoogle Scholar
  32. 32.
    Hill DL, Rao PV (1977) Tests of symmetry based on Cramer–von Mises statistics. Biometrika 64(3):489–494MATHMathSciNetGoogle Scholar
  33. 33.
    Hill DL, Rao PV (1981) Tests of symmetry based on Watson statistics. Commun Stat Theory Methods 10(11):1111–1125CrossRefMathSciNetGoogle Scholar
  34. 34.
    Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Ann Math Stat 3(2):141–142CrossRefGoogle Scholar
  35. 35.
    Huber PJ (1972) Robust statistics: A review. Ann Math Stat 43:1041–1067MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    John JA, Draper NR (1980) An alternative family of transformations. Appl Stat 29(2):190–197MATHCrossRefGoogle Scholar
  37. 37.
    Koopman PAR (1979) Testing symmetry with a procedure combining the sign test and the sign rank test. Stat Neerl 33:137–141CrossRefGoogle Scholar
  38. 38.
    Koutrouvelis IA (1985) Distribution-free procedures for location and symmetry inference problems based on the empirical characteristic function. Scand J Stat 12(4):257–269MATHMathSciNetGoogle Scholar
  39. 39.
    Koziol JA (1980) On a Cramér–von Mises-type statistic for testing symmetry. J Am Stat Assoc 75(369):161–167MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Koziol JA (1985) A note on testing symmetry with estimated parameters. Stat Probab Lett 3:227–230MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Lockhart RA, McLaren CG (1985) Asymptotic points for a test of symmetry about a specified median. Biometrika 72(1):208–210CrossRefMathSciNetGoogle Scholar
  42. 42.
    McWilliams TP (1990) A distribution-free test for symmetry based on a runs statistic. J Am Stat Assoc 85(412):1130–1133CrossRefMathSciNetGoogle Scholar
  43. 43.
    Miao W, Gel YR, Gastwirth JL (2006) A new test of symmetry about an unknown median. In: Random walk, sequential analysis and related topics: A festschrift in honor of Yuan-Shih Chow. World Scientific, New Jersey, pp 199–214Google Scholar
  44. 44.
    Mira A (1999) Distribution-free test for symmetry based on Bonferroni’s measure. J Appl Stat 26(8):959–972MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Modarres R, Gastwirth JL (1996) A modified runs test for symmetry. Stat Probab Lett 31: 107–112MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Modarres R, Gastwirth JL (1998) Hybrid test for the hypothesis of symmetry. J Appl Stat 25(6):777–783MATHCrossRefGoogle Scholar
  47. 47.
    Orlov AI (1972) On testing the symmetry of distributions. Theory Prob Appl 17:357–361MATHCrossRefGoogle Scholar
  48. 48.
    Psaradakis Z (2003) A bootstrap test for symmetry of dependent data based on a Kolmogorov–Smirnov type statistic. Comm Stat Simul Comput 32(1):113–126MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Ramberg JS, Schmeiser BW (1974) An approximate method for generating asymmetric random variables. Commun ACM 17:78–82MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Randles RH, Fligner MA, Policello II GE, Wolfe DA (1980) An asymptotically distribution-free test for symmetry versus asymmetry. J Am Stat Assoc 75(369):168–172MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Reynolds MR Jr (1975) A sequential signed-rank test for symmetry. Ann Stat 3(2):382–400MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Rosenstein RB (1989) Tests of symmetry derived as components of Pearson’s phi-squared distance measure. Commun Stat Theory Methods 18(5):1617–1626MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Rothman ED, Woodroofe M (1972) A Cramér von–Mises type statistic for testing symmetry. Ann Math Stat 43(6):2035–2038MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Schuster EF, Barker RC (1987) Using the bootstrap in testing symmetry versus asymmetry. Commun Stat Simul Comput 16(1):69–84MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, LondonMATHGoogle Scholar
  56. 56.
    Srinivasan R, Godio LB (1974) A Cramer–von Mises type statistic for testing symmetry. Biometrika 61(1):196–198MATHMathSciNetGoogle Scholar
  57. 57.
    Stuart A, Ord JK (1994) Kendall’s advanced theory of statistics, vol 1, 6 edn. Halsted, New YorkGoogle Scholar
  58. 58.
    Tajuddin IH (1994) Disribution-free test for symmetry based on Wilcoxon two-sample test. J Appl Stat 21(5):409–415CrossRefGoogle Scholar
  59. 59.
    van Eeden C, Bernard A (1957) A general class of distribution-free tests for symmetry containing the tests of Wilcoxon and Fisher. Nederl Akad Wetensch A60:381–408Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Nevada at Las VegasLas VegasUSA

Personalised recommendations