Skip to main content
SpringerLink
Log in

Construction of simultaneous confidence bands for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region

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

Abstract

This paper provides the method of constructing the simultaneous confidence band for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region. The band is compared with that on a rectangular region. It is shown that the band on the ellipsoidal region has a big advantage. The proposed method allows to construct the Type I band and the Type II band, and also allows to constrain the predictor variables either in finite intervals or in infinite intervals. In order to evaluate the efficiency of the band, the average width criterion is introduced, under which the Type I band and the Type II band are compared. All stated above are illustrated by a real example of a leaching rate study.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Al-Saidy OM, Piegorsch WW, West RW, Nitcheva DK (2003) Confidence bands for low-dose risk estimation with quantal response data. Biometrics 59:1056–1062

    Article  MathSciNet  Google Scholar 

  • Amo-Salas M, Delgado-Márquez E, Filová L, López-Fidalgo J (2016) Optimal designs for model discrimination and fitting for the flow of particles. Stat Pap 57(4):1–17

    Article  MathSciNet  Google Scholar 

  • Bowden DC (1970) Simultaneous confidence bands for linear regression models. J Am Stat Assoc 65:413–421

    Article  MathSciNet  Google Scholar 

  • Casella G, Strawderman WE (1980) Confidence bands for linear regression with restricted predictor variable. J Am Stat Assoc 75(372):862–868

    MathSciNet  MATH  Google Scholar 

  • Draper NR, Pukelsheim F (1996) An overview of design of experiments. Stat Pap 37:1–32

    Article  MathSciNet  Google Scholar 

  • Edwards D, Berry JJ (1987) The efficiency of simulation-based multiple comparisons. Biometrics 43:913–928

    Article  MathSciNet  Google Scholar 

  • Gafarian AV (1964) Confidence bands in straight line regression. J Am Stat Assoc 59:182–213

    Article  MathSciNet  Google Scholar 

  • Han Y, Liu W, Bretz F, Wan F (2015) Simultaneous confidence bands for a percentile line in linear regression. Comput Stat Data Anal 81:1–9

    Article  MathSciNet  Google Scholar 

  • Liu W (2010) Simultaneous inference in regression. CRC Press, Boca Raton, pp 70–72

    Book  Google Scholar 

  • Liu W, Ah-kine P, Zhou S (2011) Simultaneous confidence bands for linear regression with covariates constrained in intervals. Scand J Stat 39:543–553

    Article  MathSciNet  Google Scholar 

  • Liu W, Bretz F, Hayter AJ, Wynn HP (2009) Assessing nonsuperiority, noninferiority or equivalence when comparing two regression models over a restricted covariate region. Biometrics 65:1279–1287

    Article  MathSciNet  Google Scholar 

  • Liu W, Hayter AJ (2007) Minimum area confidence set optimality for confidence bands in simple linear regression. J Am Stat Assoc 102:181–190

    Article  MathSciNet  Google Scholar 

  • Liu W, Hsu JC, Bretz F, Hayter AJ, Han Y (2014) Shelf-life and its estimation in drug stability studies. J Appl Stat 41(9):1989–2000

    Article  MathSciNet  Google Scholar 

  • Liu W, Jamshidian M, Zhang Y (2004) Multiple comparison of several linear regression models. J Am Stat Assoc 99:395–403

    Article  MathSciNet  Google Scholar 

  • Liu W, Jamshidian M, Zhang Y, Donnelly J (2005) Simulation-based simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals. J Comput Graph Stat 14:459–484

    Article  MathSciNet  Google Scholar 

  • Liu W, Lin S (2009) Construction of exact simultaneous confidence bands in multiple linear regression with predictor variables constrained in an ellipsoidal region. Statistica Sinica 19:213–232

    MathSciNet  MATH  Google Scholar 

  • Naiman DQ (1984) Average width optimality of simultaneous confidence bands. Ann Stat 12:1199–1214

    Article  Google Scholar 

  • Piegorsch WW, West RW, Pan W, Kodell R (2005) Low dose risk estimation via simultaneous statistical inferences. J R Stat Soc 54:245–258

    Article  MathSciNet  Google Scholar 

  • Steinhorst RK, Bowden DC (1971) Discrimination and confidence bands on percentiles. J Am Stat Assoc 66:851–854

    Article  Google Scholar 

  • Thomas DL, Thomas DR (1986) Confidence bands for percentiles in the linear regression model. J Am Stat Assoc 81:705–708

    Article  MathSciNet  Google Scholar 

  • Turner DL, Bowden DC (1977) Simultaneous confidence bands for percentile lines in the general linear model. J Am Stat Assoc 72:886–889

    MathSciNet  MATH  Google Scholar 

  • Wang D, Li K, Xie J (1997) Optimization of technological conditions for Shuang Ju. China Pharm 2:96

    Google Scholar 

  • Xu X, Shang X (2017) D-optimal designs for full and reduced Fourier regression models. Stat Pap 58(3):811–829

    Article  MathSciNet  Google Scholar 

  • Zhou S (2017) One-sided hyperbolic simultaneous confidence bands for multiple and polynomial regression models. Metrika 80:187–200

    Article  MathSciNet  Google Scholar 

  • Zhou S, Zhu J, Wang D (2017) Simultaneous confidence bands for a percentile hyper-plane with covariates constrained in a restricted range. Working paper

Download references

Author information

Authors and Affiliations

  1. Institute of Finance and Economics Research, School of Urban and Regional Science, Shanghai University of Finance and Economics, 777 Guoding Road, 200433, Shanghai, China

    Sanyu Zhou & Jingjing Zhu

  2. School of Statistics and Management, Shanghai University of Finance and Economics, 777 Guoding Road, 200433, Shanghai, China

    Defa Wang

Authors
  1. Sanyu Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Defa Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jingjing Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sanyu Zhou.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (m 9 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, S., Wang, D. & Zhu, J. Construction of simultaneous confidence bands for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region. Stat Papers 61, 1335–1346 (2020). https://doi.org/10.1007/s00362-018-0990-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-018-0990-4

Keywords

Search

Navigation