Skip to main content
SpringerLink
Log in

Goodness-of-fit tests in parametric regression based on the estimation of the error distribution

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

Abstract

Consider a heteroscedastic regression model Y=m(X)+σ(X)ε, where m(X)=E(Y|X) and σ 2(X)=Var (Y|X) are unknown, and the error ε is independent of the covariate X. We propose a new type of test statistic for testing whether the regression curve m(⋅) belongs to some parametric family of regression functions. The proposed test statistic measures the distance between the empirical distribution function of the parametric and of the nonparametric residuals. The asymptotic theory of the proposed test is developed, and the proposed testing procedure is illustrated by means of a small simulation study and the analysis of a data set.

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

  • Akritas MG, Van Keilegom I (2001) Nonparametric estimation of the residual distribution. Scand J Stat 28:549–568

    Article  MATH  Google Scholar 

  • 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 

  • Bellver C (1987) Influence of particulate pollution on the positions of neutral points in the sky in Seville (Spain). Atmos Environ 21:699–702

    Article  Google Scholar 

  • Chen SX, Härdle W, Li M (2003) An empirical likelihood goodness-of-fit test for time series. J Roy Stat Soc Ser B 65:663–678

    Article  MATH  Google Scholar 

  • Choi E, Hall P, Rousson V (2000) Data sharpening methods for bias reduction in nonparametric regression. Ann Stat 28:1339–1355

    Article  MATH  MathSciNet  Google Scholar 

  • Cleveland WS (1993) Visualizing data. Hobart Press, Summit

    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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H, Munk A, Wagner T (2000) Testing model assumptions in multivariate linear regression models. J Nonparametric Stat 12:309–342

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J, Zhang C, Zhang J (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Ann Stat 29:153–193

    Article  MATH  MathSciNet  Google Scholar 

  • González Manteiga W, Prada Sánchez JM, Romo J (1994) The bootstrap—a review. Comput Stat 9:165–205

    Google Scholar 

  • Hall P, DiCiccio TJ, Romano JP (1989) On smoothing and the bootstrap. Ann Stat 17:692–704

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Hart JD (1997) Nonparametric smoothing and lack-of-fit tests. Springer, New York

    MATH  Google Scholar 

  • Jennrich RI (1969) Asymptotic properties of non-linear least-squares estimators. Ann Math Stat 40:633–643

    Article  MATH  MathSciNet  Google Scholar 

  • Koul HL, Lahiri SN (1994) On bootstrapping M-estimated residual processes in multiple linear regression models. J Multivar Anal 49:255–265

    Article  MATH  MathSciNet  Google Scholar 

  • Mammen E (2000) Resampling methods for nonparametric regression. In: Schimek MG (ed) Smoothing and regression. Wiley, New York

    Google Scholar 

  • Sánchez Sellero C (2001) Inferencia estadística en datos con censura y/o truncamiento. PhD thesis, University of Santiago de Compostela

  • Seber GAF, Wild CJ (1989) Nonlinear regression. Wiley, New York

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Stute W, González Manteiga W, Presedo Quindimil M (1998) Bootstrap approximations in model checks for regression. J Am Stat Assoc 93:141–149

    Article  MATH  Google Scholar 

  • Van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Van Keilegom I, Veraverbeke N (2002) Density and hazard estimation in censored regression models. Bernoulli 8:607–625

    MATH  MathSciNet  Google Scholar 

  • White H (1981) Consequences and detection of misspecified nonlinear regression models. J Am Stat Assoc 76:419–433

    Article  MATH  Google Scholar 

  • White H (1982) Maximum likelihood estimation of misspecified models. Econometrica 50:1–25

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Zhang CM (2003) Adaptive tests of regression functions via multiscale generalized likelihood ratios. Can J Stat 31:151–171

    Article  MATH  Google Scholar 

  • Zhang CM, Dette H (2004) A power comparison between nonparametric regression tests. Stat Probab Lett 66:289–301

    Article  MATH  MathSciNet  Google Scholar 

  • Zhu L-X (2003) Model checking of dimension-reduction type for regression. Stat Sin 13:283–296

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Statistique, Université Catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    Ingrid Van Keilegom

  2. Departamento de Estadística, Facultad de Matemáticas, Universidad de Santiago de Compostela, Campus Sur, 15706, Santiago de Compostela, Spain

    Wenceslao González Manteiga & César Sánchez Sellero

Authors
  1. Ingrid Van Keilegom
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenceslao González Manteiga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. César Sánchez Sellero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenceslao González Manteiga.

Additional information

I.V.K. is financially supported by the IAP research network nr. P5/24 of the Belgian government (Belgian Science Policy). W.G.M. and C.S.S. are financially supported by the Spanish Ministry of Science and Technology (with additional European FEDER support) through project MTM2005-00820 and from Xunta de Galicia through project PGIDIT03PXIC20702PN.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van Keilegom, I., González Manteiga, W. & Sánchez Sellero, C. Goodness-of-fit tests in parametric regression based on the estimation of the error distribution. TEST 17, 401–415 (2008). https://doi.org/10.1007/s11749-007-0044-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-007-0044-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation