Advertisement

Statistical Papers

, Volume 60, Issue 3, pp 963–981 | Cite as

Nonparametric tests for ordered quantiles

  • Pooja SoniEmail author
  • Isha Dewan
  • Kanchan Jain
Regular Article
  • 169 Downloads

Abstract

In this paper, nonparametric procedures for testing equality of quantiles against an ordered alternative are proposed. These testing procedures are based on two different estimators of the quantile function available in literature. Limiting distributions of the test statistics are derived. Simulations have been carried out to check the performance of the tests.

Keywords

Quantiles Kernel estimator Empirical estimator Multiple comparison procedure Censored data 

Mathematics Subject Classification

62N01 

Notes

Acknowledgements

The authors are grateful to the referees for their valuable suggestions which have contributed significantly in improving the manuscript. The third author acknowledges the support provided by Department of Science and Technology, Govt. of India under PURSE Grants.

References

  1. Bennette C, Vickers A (2012) Against quantiles: categorization of continuous variables in epidemiologic research, and its discontents. BMC Med Res Methodol 12:21–25CrossRefGoogle Scholar
  2. Brookmeyer R, Crowley J (1982) A k-sample median test for censored data. J Am Stat Assoc 77:433–440MathSciNetzbMATHGoogle Scholar
  3. Chen Z (2014) Extension of mood’s median test for survival data. Stat Probab Lett 95:77–84MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chen Z (2017) A nonparametric approach to detecting the difference of survival medians. Commun Stat Simul Comput 46(1):395–403Google Scholar
  5. Cheung S, Wu K, Lim S (2002) Simultaneous prediction intervals for multiple comparisons with a standard. Stat Pap 43:337–347MathSciNetCrossRefzbMATHGoogle Scholar
  6. Falk M (1985) Asymptotic normality of the kernel quantile estimator. Ann Stat 13:428–433MathSciNetCrossRefzbMATHGoogle Scholar
  7. Farrell PM, Kosorok MR, Laxova A, Shen G, Koscik RE, Bruns WT, Splaingard M, Mischler EH (1997) Nutritional benefits of neonatal screening for cystic fibrosis. N Engl J Med 337:963–969CrossRefGoogle Scholar
  8. Gupta SS, Panchapakesan S (1979) Multiple decision procedures: theory and methodology of selecting and ranking populations. Wiley, New YorkzbMATHGoogle Scholar
  9. Hayter AJ, Liu W (1996) Exact calculations for the one-sided studentized range test for testing against a simple ordered alternative. Comput Stat Data Anal 22:17–25MathSciNetCrossRefGoogle Scholar
  10. Hochberg Y (1976) A modification of the \(T\)-method of multiple comparisons for a one-way layout with unequal variances. J Am Stat Assoc 71:200–203MathSciNetzbMATHGoogle Scholar
  11. Hochberg Y, Tamhane AC (1987) Multiple comparison procedures. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  12. Jones M (1992) Estimating densities, quantiles, quantile densities and density quantiles. Ann Inst Stat Math 44:721–727CrossRefzbMATHGoogle Scholar
  13. Kotz S, Seier E (2009) An analysis of quantile measures of kurtosis: center and tails. Stat Pap 50:553–568MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lee RE, Spurrier JD (1995) Successive comparisons between ordered treatments. J Stat Plan Inference 43:323–330MathSciNetCrossRefzbMATHGoogle Scholar
  15. Lio YL, Padgett WJ, Yu KF (1986) On the asymptotic properties of a kernel type quantile estimator from censored samples. J Stat Plan Inference 14:169–177MathSciNetCrossRefzbMATHGoogle Scholar
  16. Miller RG Jr (1966) Simultaneous statistical inference. McGraw-Hill, New YorkzbMATHGoogle Scholar
  17. Nair NU, Sankaran P, Balakrishnan N (2013) Quantile-based reliability analysis. Birkhauser Verlag, New YorkCrossRefzbMATHGoogle Scholar
  18. Nakayama M (2007) Single-stage multiple-comparison procedure for quantiles and other parameters. In: Proceedings of the 2007 winter simulation conference, Washington, DC, pp 530–534Google Scholar
  19. Nashimoto K, Wright FT (2007) Nonparametric multiple-comparison methods for simply ordered medians. Comput Stat Data Anal 51:5068–5076MathSciNetCrossRefzbMATHGoogle Scholar
  20. Nashimoto K, Haldeman KM, Tait CM (2013) Multiple comparisons of binomial proportions. Comput Stat Data Anal 68:202–212MathSciNetCrossRefzbMATHGoogle Scholar
  21. Padgett WJ (1986) A kernel-type estimator of a quantile function from right-censored data. J Am Stat Assoc 81:215–222MathSciNetCrossRefzbMATHGoogle Scholar
  22. Parzen E (1979) Nonparametric statistical data modeling. J Am Stat Assoc 74:105–131MathSciNetCrossRefzbMATHGoogle Scholar
  23. Rahbar MH, Chen Z, Jeon S, Gardiner JC, Ning J (2012) A nonparametric test for equality of survival medians. Stat Med 31:844–854MathSciNetCrossRefGoogle Scholar
  24. Sander JM (1975) The weak convergence of quantiles of the product-limit estimator. Stanford University, StanfordGoogle Scholar
  25. Sankaran PG, Midhu NN (2016) Testing exponentiality using mean residual quantile function. Stat Pap 57:235–247MathSciNetCrossRefzbMATHGoogle Scholar
  26. Scheffé H (1953) A method for judging all contrasts in the analysis of variance. Biometrika 40:87–104MathSciNetzbMATHGoogle Scholar
  27. Scheffé H (1956) A “mixed model” for the analysis of variance. Ann Math Stat 27:23–36MathSciNetCrossRefzbMATHGoogle Scholar
  28. Scheffé H (1959) Anal Var. Wiley, New YorkGoogle Scholar
  29. Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  30. Shen J, He S (2007) Empirical likelihood for the difference of quantiles under censorship. Stat Pap 48:437–457MathSciNetCrossRefzbMATHGoogle Scholar
  31. Singh P, Gill AN (2002) Multiple comparison of ranked location parameters: logistic populations case. J Comb Inf Syst Sci 27:173–183MathSciNetzbMATHGoogle Scholar
  32. Somerville P (2001) Combining one-sided and two-sided confidence interval procedures for successive comparisons of ordered treatment effects. Biom J 43:533–542MathSciNetCrossRefzbMATHGoogle Scholar
  33. Soni P, Dewan I, Jain K (2012) Nonparametric estimation of quantile density function. Comput Stat Data Anal 56:3876–3886MathSciNetCrossRefzbMATHGoogle Scholar
  34. Soni P, Dewan I, Jain K (2015) Tests for successive differences of quantiles. Stat Probab Lett 97:1–8MathSciNetCrossRefzbMATHGoogle Scholar
  35. Stevens G (1989) Multiple comparison test for differences in scale parameters. Metrika 36:91–106MathSciNetCrossRefzbMATHGoogle Scholar
  36. Tamhane AC (1977) Multiple comparisons in model I one-way ANOVA with unequal variances. Commun Stat Theory Methods 6:15–32MathSciNetCrossRefzbMATHGoogle Scholar
  37. Tamhane AC (1979) A comparison of procedures for multiple comparisons of means with unequal variances. J Am Stat Assoc 74:471–480MathSciNetzbMATHGoogle Scholar
  38. Tukey JW (1949) Comparing individual means in the analysis of variance. Biometrics 5:99–114MathSciNetCrossRefGoogle Scholar
  39. Wu SF, Chen HJ (1997) Multiple comparison procedures with the average for exponential location parameters when sample sizes are unequal. Int J Inf Manag Sci 8:41–54MathSciNetzbMATHGoogle Scholar
  40. Wu SF, Chen HJ (1999) Multiple comparison procedures with the average for scale families: normal and exponential populations. Commun Stat Simul Comput 28:73–98MathSciNetCrossRefzbMATHGoogle Scholar
  41. Yang SS (1985) A smooth nonparametric estimator of a quantile function. J Am Stat Assoc 80:1004–1011MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.University Business SchoolPanjab UniversityChandigarhIndia
  2. 2.Theoretical Statistics and Mathematics UnitIndian Statistical InstituteNew DelhiIndia
  3. 3.Department of StatisticsPanjab UniversityChandigarhIndia

Personalised recommendations