Skip to main content
SpringerLink
Log in

Nonparametric tests for ordered quantiles

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

A Publisher Correction to this article was published on 12 September 2019

This article has been updated

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 12 September 2019

    Unfortunately, due to a technical error, the articles published in issues 60:2 and 60:3 received incorrect pagination. Please find here the corrected Tables of Contents. We apologize to the authors of the articles and the readers.

References

  • Bennette C, Vickers A (2012) Against quantiles: categorization of continuous variables in epidemiologic research, and its discontents. BMC Med Res Methodol 12:21–25

    Article  Google Scholar 

  • Brookmeyer R, Crowley J (1982) A k-sample median test for censored data. J Am Stat Assoc 77:433–440

    MathSciNet  MATH  Google Scholar 

  • Chen Z (2014) Extension of mood’s median test for survival data. Stat Probab Lett 95:77–84

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Z (2017) A nonparametric approach to detecting the difference of survival medians. Commun Stat Simul Comput 46(1):395–403

  • Cheung S, Wu K, Lim S (2002) Simultaneous prediction intervals for multiple comparisons with a standard. Stat Pap 43:337–347

    Article  MathSciNet  MATH  Google Scholar 

  • Falk M (1985) Asymptotic normality of the kernel quantile estimator. Ann Stat 13:428–433

    Article  MathSciNet  MATH  Google Scholar 

  • 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 337:963–969

    Article  Google Scholar 

  • Gupta SS, Panchapakesan S (1979) Multiple decision procedures: theory and methodology of selecting and ranking populations. Wiley, New York

    MATH  Google Scholar 

  • 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 22:17–25

    Article  MathSciNet  Google Scholar 

  • Hochberg Y (1976) A modification of the \(T\)-method of multiple comparisons for a one-way layout with unequal variances. J Am Stat Assoc 71:200–203

    MathSciNet  MATH  Google Scholar 

  • Hochberg Y, Tamhane AC (1987) Multiple comparison procedures. Wiley, New York

    Book  MATH  Google Scholar 

  • Jones M (1992) Estimating densities, quantiles, quantile densities and density quantiles. Ann Inst Stat Math 44:721–727

    Article  MATH  Google Scholar 

  • Kotz S, Seier E (2009) An analysis of quantile measures of kurtosis: center and tails. Stat Pap 50:553–568

    Article  MathSciNet  MATH  Google Scholar 

  • Lee RE, Spurrier JD (1995) Successive comparisons between ordered treatments. J Stat Plan Inference 43:323–330

    Article  MathSciNet  MATH  Google Scholar 

  • Lio YL, Padgett WJ, Yu KF (1986) On the asymptotic properties of a kernel type quantile estimator from censored samples. J Stat Plan Inference 14:169–177

    Article  MathSciNet  MATH  Google Scholar 

  • Miller RG Jr (1966) Simultaneous statistical inference. McGraw-Hill, New York

    MATH  Google Scholar 

  • Nair NU, Sankaran P, Balakrishnan N (2013) Quantile-based reliability analysis. Birkhauser Verlag, New York

    Book  MATH  Google Scholar 

  • Nakayama M (2007) Single-stage multiple-comparison procedure for quantiles and other parameters. In: Proceedings of the 2007 winter simulation conference, Washington, DC, pp 530–534

  • Nashimoto K, Wright FT (2007) Nonparametric multiple-comparison methods for simply ordered medians. Comput Stat Data Anal 51:5068–5076

    Article  MathSciNet  MATH  Google Scholar 

  • Nashimoto K, Haldeman KM, Tait CM (2013) Multiple comparisons of binomial proportions. Comput Stat Data Anal 68:202–212

    Article  MathSciNet  MATH  Google Scholar 

  • Padgett WJ (1986) A kernel-type estimator of a quantile function from right-censored data. J Am Stat Assoc 81:215–222

    Article  MathSciNet  MATH  Google Scholar 

  • Parzen E (1979) Nonparametric statistical data modeling. J Am Stat Assoc 74:105–131

    Article  MathSciNet  MATH  Google Scholar 

  • Rahbar MH, Chen Z, Jeon S, Gardiner JC, Ning J (2012) A nonparametric test for equality of survival medians. Stat Med 31:844–854

    Article  MathSciNet  Google Scholar 

  • Sander JM (1975) The weak convergence of quantiles of the product-limit estimator. Stanford University, Stanford

    Google Scholar 

  • Sankaran PG, Midhu NN (2016) Testing exponentiality using mean residual quantile function. Stat Pap 57:235–247

    Article  MathSciNet  MATH  Google Scholar 

  • Scheffé H (1953) A method for judging all contrasts in the analysis of variance. Biometrika 40:87–104

    MathSciNet  MATH  Google Scholar 

  • Scheffé H (1956) A “mixed model” for the analysis of variance. Ann Math Stat 27:23–36

    Article  MathSciNet  MATH  Google Scholar 

  • Scheffé H (1959) Anal Var. Wiley, New York

    Google Scholar 

  • Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York

    Book  MATH  Google Scholar 

  • Shen J, He S (2007) Empirical likelihood for the difference of quantiles under censorship. Stat Pap 48:437–457

    Article  MathSciNet  MATH  Google Scholar 

  • Singh P, Gill AN (2002) Multiple comparison of ranked location parameters: logistic populations case. J Comb Inf Syst Sci 27:173–183

    MathSciNet  MATH  Google Scholar 

  • Somerville P (2001) Combining one-sided and two-sided confidence interval procedures for successive comparisons of ordered treatment effects. Biom J 43:533–542

    Article  MathSciNet  MATH  Google Scholar 

  • Soni P, Dewan I, Jain K (2012) Nonparametric estimation of quantile density function. Comput Stat Data Anal 56:3876–3886

    Article  MathSciNet  MATH  Google Scholar 

  • Soni P, Dewan I, Jain K (2015) Tests for successive differences of quantiles. Stat Probab Lett 97:1–8

    Article  MathSciNet  MATH  Google Scholar 

  • Stevens G (1989) Multiple comparison test for differences in scale parameters. Metrika 36:91–106

    Article  MathSciNet  MATH  Google Scholar 

  • Tamhane AC (1977) Multiple comparisons in model I one-way ANOVA with unequal variances. Commun Stat Theory Methods 6:15–32

    Article  MathSciNet  MATH  Google Scholar 

  • Tamhane AC (1979) A comparison of procedures for multiple comparisons of means with unequal variances. J Am Stat Assoc 74:471–480

    MathSciNet  MATH  Google Scholar 

  • Tukey JW (1949) Comparing individual means in the analysis of variance. Biometrics 5:99–114

    Article  MathSciNet  Google Scholar 

  • 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 8:41–54

    MathSciNet  MATH  Google Scholar 

  • Wu SF, Chen HJ (1999) Multiple comparison procedures with the average for scale families: normal and exponential populations. Commun Stat Simul Comput 28:73–98

    Article  MathSciNet  MATH  Google Scholar 

  • Yang SS (1985) A smooth nonparametric estimator of a quantile function. J Am Stat Assoc 80:1004–1011

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. University Business School, Panjab University, Chandigarh, 160014, India

    Pooja Soni

  2. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, New Delhi, 110016, India

    Isha Dewan

  3. Department of Statistics, Panjab University, Chandigarh, 160014, India

    Kanchan Jain

Authors
  1. Pooja Soni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Isha Dewan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kanchan Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pooja Soni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Tables 12 and 13 show the powers when the chosen distribution is exponential with parameter \(\lambda _i = 1+(i-1)0.2\), for \(i=1,2,\ldots ,k\) and \(\mathrm{k}\,=\,3\) and 4 respectively.

Table 12 Power for Exp(1), Exp(5/6), Exp(5/7)
Full size table
Table 13 Power for Exp(1), Exp(5/6), Exp(5/7), Exp(5/8)
Full size table

Tables 14 and 15 depict power of the tests for \(k = 3\) and 4 when underlying distribution is Weibull with increasing failure rate i.e. with same shape (\(\alpha >1\)) but different scale parameters (\(\lambda _i\), for \(i=1,2,\ldots ,k\)). The ordering of scale parameters of Weibull distributions leads to ordering of their quantiles .

Table 14 Power for Weibull(2,1), Weibull(2,1.2), Weibull(2,1.4)
Full size table
Table 15 Power for Weibull(2,1), Weibull(2,1.2), Weibull(2,1.4), Weibull(2,1.6)
Full size table

Tables 16 and 17 display power of tests for underlying distribution as GLD when k \(=\) 3 and 4 respectively. \(\lambda _1\)(location parameter) is varied in Tables 16 and 17, while other parameters are kept fixed. Ordering of quantiles is maintained by varying location.

Table 16 Power for GLD(3.6,1,2,1), GLD(3.8,1,2,1), GLD(4,1,2,1)
Full size table
Table 17 Power for GLD(3.4,1,2,1), GLD(3.6,1,2,1), GLD(3.8,1,2,1), GLD(4,1,2,1)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soni, P., Dewan, I. & Jain, K. Nonparametric tests for ordered quantiles. Stat Papers 60, 963–981 (2019). https://doi.org/10.1007/s00362-016-0859-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-016-0859-3

Keywords

Mathematics Subject Classification

Search

Navigation