Skip to main content
SpringerLink
Log in

Model checks for parametric regression models

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

In the paper we deal with the problem of model selection among fixed-design regression models. We establish a new test that indicates whether or not the model fits the data. The test statistic is based on the difference between a parametric estimator for the model variance and a nonparametric difference-based estimator, see Hall et al. (Biometrika 77:521–528, 1990). The weights in the nonparametric estimator depend on n, and they are chosen by solving an optimisation problem in order to obtain a test with high power.

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

References

  • Alcalá JT, Cristóbal JA, González-Manteiga W (1999) Goodness-of-fit test for linear models based on local polynomials. Stat Probab Lett 42:39–46

    Article  MATH  Google Scholar 

  • Brodeau F (1993) Tests for the choice of approximative models in nonlinear regression when the variance is unknown. Statistics 24:95–106

    Article  MathSciNet  MATH  Google Scholar 

  • Brown LD, Levine M (2007) Variance estimation in nonparametric regression via the difference sequence method. Ann Stat 35:2219–2232

    Article  MathSciNet  MATH  Google Scholar 

  • Dette H (1999) A consistent test for the functional form of a regression based on a difference of variance estimators. Ann Stat 27:1012–1040

    Article  MathSciNet  MATH  Google Scholar 

  • Dette H (2002) A consistent test for heteroscedasticity in nonparametric regression based on kernel method. J Stat Plan Inference 103:311–329

    Article  MathSciNet  MATH  Google Scholar 

  • Dette H, Munk A (1998) Validation of linear regression models. Ann Stat 26:778–800

    Article  MathSciNet  MATH  Google Scholar 

  • Dette H, Munk A, Wagner T (1998) Estimating the variance in nonparametric regression: What is a reasonable choice? J R Stat Soc B 60:751–764

    Article  MathSciNet  MATH  Google Scholar 

  • Du J, Schick A (2008) Covariate-matched estimator of the error variance in nonparametric regression. Working paper

  • Eubank RL, Hart JD (1992) Testing goodness-of-fit in regression via order selection criteria. Ann Stat 20:1412–1425

    Article  MathSciNet  MATH  Google Scholar 

  • Gasser T, Sroka L, Jennen-Steinmetz C (1986) Residual variance and residual pattern in nonlinear regression. Biometrika 73:625–633

    Article  MathSciNet  MATH  Google Scholar 

  • Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947

    Article  MATH  Google Scholar 

  • Hall P, Heyde CC (1980) Martingale limit theory and its application. Academic Press, San Diego

    MATH  Google Scholar 

  • Hall P, Marron JS (1990) On variance estimation in nonparametric regression. Biometrika 77:415–419

    Article  MathSciNet  MATH  Google Scholar 

  • Hall P, Kay JW, Titterington DM (1990) Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77:521–528

    Article  MathSciNet  Google Scholar 

  • Horowitz JL, Spokoiny VG (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69:599–631

    Article  MathSciNet  MATH  Google Scholar 

  • Lachout P, Liebscher E, Vogel S (2005) Strong convergence of estimators as ε n -minimisers of optimisation problems. Ann Inst Stat Math 57:291–313

    Article  MathSciNet  MATH  Google Scholar 

  • Müller H-G, Stadtmüller U (1993) On variance estimation with quadratic forms. J Stat Plan Inference 35:213–231

    Article  MATH  Google Scholar 

  • Müller H-G, Zhao P-L (1995) On a semiparametric variance function model and a test for heteroscedasticity. Ann Stat 23:946–967

    Article  MATH  Google Scholar 

  • Müller UU, Schick A, Wefelmeyer W (2003) Estimating the error variance in nonparametric regression by a covariate-matched U-statistic. Statistics 37:179–188

    Article  MathSciNet  MATH  Google Scholar 

  • Munk A, Bissantz N, Wagner T, Freitag G (2005) On difference based variance estimation in nonparametric regression when the covariate is high dimensional. J R Stat Soc B 67:19–41

    Article  MathSciNet  MATH  Google Scholar 

  • Neumann M (1994) Fully data-driven nonparametric variance estimators. Statistics 25:189–212

    Article  MathSciNet  MATH  Google Scholar 

  • Rice JA (1984) Bandwidth choice for nonparametric regression. Ann Stat 12:1215–1230

    Article  MathSciNet  MATH  Google Scholar 

  • Ruppert D, Wand MP, Holst U, Hössjer O (1997) Local polynomial variance-function estimation. Technometrics 39:262–273

    Article  MathSciNet  MATH  Google Scholar 

  • Sheehy A, Gasser T, Rousson V (2005) Nonparametric measures of variance explained. J Nonparametr Stat 17:765–776

    Article  MathSciNet  MATH  Google Scholar 

  • Stute W (1997) Nonparametric model checks for regression. Ann Stat 25:613–641

    Article  MathSciNet  MATH  Google Scholar 

  • Stute W, Thies S, Zhu L-X (1998) Model checks for regression: An innovation process approach. Ann Stat 26:1916–1934

    Article  MathSciNet  MATH  Google Scholar 

  • Tong T, Wang Y (2005) Estimating residual variance in nonparametric regression using least squares. Biometrika 92:821–830

    Article  MathSciNet  MATH  Google Scholar 

  • Tong T, Liu A, Wang Y (2008) Relative errors of difference-based variance estimators in nonparametric regression. Commun Stat, Theory Methods 37:2890–2902

    Article  MathSciNet  Google Scholar 

  • Wu C-F (1981) Asymptotic theory of nonlinear least squares estimation. Ann Stat 9:501–513

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Communication Systems, University of Applied Sciences Merseburg, Geusaer Straße, 06217, Merseburg, Germany

    Eckhard Liebscher

Authors
  1. Eckhard Liebscher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eckhard Liebscher.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liebscher, E. Model checks for parametric regression models. TEST 21, 132–155 (2012). https://doi.org/10.1007/s11749-011-0239-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-011-0239-1

Keywords

Mathematics Subject Classification

Search

Navigation