The Annals of Statistics

Significance testing in nonparametric regression based on the bootstrap

Miguel A. Delgado and Wenceslao González Manteiga

Full-text: Open access

Abstract

This paper proposes a test for selecting explanatory variables in nonparametric regression. The test does not need to estimate the conditional expectation function given all the variables, but only those which are significant under the null hypothesis. This feature is computationally convenient and solves, in part, the problem of the “curse of dimensionality” when selecting regressors in a nonparametric context. The proposed test statistic is based on functionals of a $U$-process. Contiguous alternatives, converging to the null at a rate $n^{-1/2}$ can be detected. The asymptotic null distribution of the statistic depends on certain features of the data generating process,and asymptotic tests are difficult to implement except in rare circumstances. We justify the consistency of two easy to implement bootstrap tests which exhibit good level accuracy for fairly small samples, according to the reported Monte Carlo simulations. These results are also applicable to test other interesting restrictions on nonparametric curves, like partial linearity and conditional independence.

Article information

Source
Ann. Statist. Volume 29, Number 5 (2001), 1469-1507.

Dates
First available in Project Euclid: 8 February 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1013203462

Digital Object Identifier
doi:10.1214/aos/1013203462

Mathematical Reviews number (MathSciNet)
MR1873339

Zentralblatt MATH identifier
1043.62032

Subjects
Primary: 62G07: Density estimation 62G09: Resampling methods 62G10: Hypothesis testing

Keywords
Nonparametric regression selection of variables higher order kernels U-processes wild bootstrap restrictions on nonparametric curves

Citation

Delgado, Miguel A.; Manteiga, Wenceslao González. Significance testing in nonparametric regression based on the bootstrap. Ann. Statist. 29 (2001), no. 5, 1469--1507. doi:10.1214/aos/1013203462. https://projecteuclid.org/euclid.aos/1013203462


Export citation

References

  • Beran, R., Le Cam, L. and Millar, P.W. (1987). Convergence of stochastic empirical measures. J. Multivariate Anal. 23 159-168.
  • Bierens, H. and Ploberger, W. (1997). Asymptotic theory of integrated conditional moment test. Econometrica 65 1153-1174.
  • Brunk, H.D. (1970). Estimation by isotonic regression. In Nonparametric Techniques in Statistical Inference (M.L. Puri ed.) 177-197. Cambridge Univ. Press.
  • Duddley, R.M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
  • de la Pe na, V.H. and Gin´e, E. (1999). Decoupling: From Dependence to Independence. Springer, Berlin.
  • Eubank, R. and Spiegelman, S. (1990). Testing the goodness of fit of a linear model via nonparametric regression techniques, J. Amer. Statist. Assoc. 85 387-392.
  • Fan, Y. and Li, Q. (1996). Consistent model specification tests: omitted variables and semiparametric functional forms. Econometrica 64 865-890.
  • Ghosal, S., Sen, A. and Van der Vaart, A. W. (2000). Testing monotonicity of regression. Ann. Statist. 28 1054-1082.
  • Gin´e, E. (1997). Lectures on Some Aspects of the Bootstrap. Ecole de ´Ete de Calcul de Probabilit´es de Saint-Flour. Lecture Notes in Math. 1665. Springer, Berlin. (See also www.math.uconn.edu/ gine/Corrections.)
  • H¨ardle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926-1947.
  • Hart, J.D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, Berlin.
  • Heckman, N.E. (1986). Spline smoothing in a partly linear model. J. Roy. Statist. Soc. Ser. B 48 244-248.
  • Hoffman-Jørgensen, J. (1984). Stochastic processes on Polish spaces. [Published (1991). Various Publication Series No. 39. Matematisk Institute, Aarhus Univ.]
  • Hong-zhy, A. and Bing, C. (1991). A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series. Internat. Statist. Rev. 59 287-307.
  • Koul, H.L. and Stute, W. (1999). Nonparametric model checks for time series. Ann. Statist. 27 204-236.
  • Ledoux, M. and Talagrand, M. (1988). Un crit ere sur les petites boules dans le th´eor eme limite central. Probab. Theory Related Fields 77 29-47.
  • Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence. Ann. Statist. 15 780-799.
  • Robinson, P.M. (1988). Root-n-consistent semiparametric regression. Econometrica 56 931-954.
  • Rosenblatt, M. (1975). A quadratic measure of deviations of two-dimensional density estimates an a test of independence. Ann. Statist. 3 1-14.
  • Sherman, R.P. (1994). Maximal inequalities for degenerate U-processes with applications to optimization estimators. Ann. Statist. 22 439-459.
  • Speckman, P. (1988). Kernel smoothing in partially linear models. J. Roy. Statist. Soc. Ser. B 50 413-446.
  • Stute, W. (1994). U-Statistic processes: a martingale approach. Ann. Probab. 22 1725-1744.
  • Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641.
  • Stute, W., Gonz´alez-Manteiga, W. and Presedo-Quindimil, M. (1998). Bootstrap approximations in model checks for regression. J. Amer. Statist. Assoc. 93 141-149.
  • Stute, W., Thies, S. and Zhu, L.X. (1998). Model checks for regression: an innovation process approach. Ann. Statist. 26 1916-1934.
  • Sue, J.Q. and Wei, L.J. (1991). A lack of fit test for the mean function in a generalized linear model. J. Amer. Statist. Assoc. 86 420-426.
  • Van der Vaart, A.W. (1994). Weak convergence of smoothed empirical processes. Scand. J. Statist. 21 501-504.
  • Van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence of Empirical Processes. Springer, New York.
  • Wu, C.F.J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. Ann. Statist. 14 1261-1350.