On Kernel-Based Mode Estimation Using Different Stratified Sampling Designs


In the literature, the properties and the application of mode estimation is considered under simple random sampling and ranked set sampling (RSS). We investigate some of the asymptotic properties of kernel density-based mode estimation using stratified simple random sampling (SSRS) and stratified ranked set sampling designs (SRSS). We demonstrate that kernel density-based mode estimation using SRSS and SSRS is consistent, asymptotically normally distributed and using SRSS has smaller variance than that under SSRS. Improved performance of the mode estimation using SRSS compared to SSRS is supported through a simulation study. We will illustrate the method by using biomarker data collected in China Health and Nutrition Survey data.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    Bellhouse DR, Stafford JE (1999) Density estimation from complex surveys. Stat Sin 9:407–424

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Breunig R (2008) Nonparametric density estimation for stratified samples. Stat Probab Lett 78:2194–2200

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Buch-Larsen T, Nielsen JP, Guillen M, Bolance C (2005) Kernel density estimation for heavy-tailed distributions using the champernowne transformation. Statistics 39(6):503–518

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Chen Z (1999) Density estimation using ranked-set sampling data. Environ Ecol Stat 6(2):135–146

    MathSciNet  Google Scholar 

  5. 5.

    Chen Z (2007) Ranked set sampling: Its essence and new applications. Environ Ecol Stat 14:355–363

    MathSciNet  Google Scholar 

  6. 6.

    Härdle W (2004) Nonparametric and semiparametric models. Springer, Berlin

    MATH  Google Scholar 

  7. 7.

    Hedges SB, Shah P (2003) Comparison of mode estimation methods and application in molecular clock analysis. BMC Bioinform 4(1):31

    Google Scholar 

  8. 8.

    Jabrah R, Samawi H, Vogel R, Rochani H, Linder D, Klibert J (2017) Using ranked auxiliary covariate as a more efficient sampling design for ANCOVA model: analysis of a psychological intervention to buttress resilience. Commun Stat Appl Methods 24(3):241–257

    Google Scholar 

  9. 9.

    Jeffrey SS (1996) Smoothing methods in statistics. Springer, New York

    MATH  Google Scholar 

  10. 10.

    Kaur A, Patil G, Sinha A, Taillie C (1995) Ranked set sampling: an annotated bibliography. Environ Ecol Stat 2(1):25–54

    Google Scholar 

  11. 11.

    Kim J, Scott CD (2012) Robust kernel density estimation. J Mach Learn Res 13:2529–2565

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Lim J, Chen M, Park S, Wang X, Stokes L (2014) Kernel density estimator from ranked set samples. Commun Stat-Theory Methods 43:2156–2168

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mahdizadeh M, Zamanzade E (2016) Kernel-based estimation of P(X > Y) in ranked set sampling. Stat Oper Res Trans (SORT) 40(2):243–266

    MathSciNet  MATH  Google Scholar 

  14. 14.

    McIntyre GA (1952) A method for unbiased selective sampling, using ranked sets. Aust J Agric Res 3:90–385

    Google Scholar 

  15. 15.

    Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33:1065–1076

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Samawi HM (1996) Stratified ranked set sample. Pak J Stat 12(1):9–16

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Samawi HM, Al-Sageer OA (2001) On the estimation of the distribution function using extreme and median ranked set sampling. Biom J 43(3):357–373

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Samawi HM, Chatterjee A, Yin J, Rochani H (2017) On kernel density estimation based on different stratified sampling. Commun Stat Theory Methods 46(22):10973–10990

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Samawi H, Rochani H, Yin J, Linder D, Vogel R (2018) Notes on kernel density based mode estimation using more efficient sampling designs. Comput Stat 33(2):1071–1090

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Silverman BW (1986) Density estimation for statistics and data analysis, vol 26. CRC Press, Boca Raton

    MATH  Google Scholar 

  21. 21.

    Takahasi K, Wakimoto K (1968) On unbiased estimates of the population mean based on the sample stratified by means of ordering. Ann Inst Stat Math 20(1):1–31

    MATH  Google Scholar 

  22. 22.

    Wand M, Jones M (1995) Kernel Smoothing of Monographs on statistics and applied probability, vol 60. Chapman and Hall, London

    Google Scholar 

  23. 23.

    Yan S, Li J, Li S, Zhang B, Du S, Gordon-Larsen P, Popkin B (2012) The expanding burden of cardiometabolic risk in China: the China Health and Nutrition Survey. Obes Rev 13(9):810–821

    Google Scholar 

Download references


The authors thank the associate editor and the reviewers for their valuable comments which improve the manuscript.

Author information



Corresponding author

Correspondence to Hani Samawi.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Samawi, H., Rochani, H., Yin, J. et al. On Kernel-Based Mode Estimation Using Different Stratified Sampling Designs. J Stat Theory Pract 13, 30 (2019). https://doi.org/10.1007/s42519-018-0034-3

Download citation


  • Mode estimation
  • Density kernel estimation
  • Stratified ranked set sampling
  • Stratified simple random sample
  • China health and nutrition survey