The Annals of Statistics

Local Asymptotics for Regression Splines and Confidence Regions

X. Shen, D.A. Wolfe, and S. Zhou

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Abstract

In this paper, we study the local behavior of regression splines. In particular, explicit expressions for the asymptotic pointwise bias and variance of regression splines are obtained. In addition, asymptotic normality for regression splines is established, leading to the construction of approximate confidence intervals and confidence bands for the regression function.

Article information

Source
Ann. Statist., Volume 26, Number 5 (1998), 1760-1782.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1024691356

Digital Object Identifier
doi:10.1214/aos/1024691356

Mathematical Reviews number (MathSciNet)
MR1673277

Zentralblatt MATH identifier
0929.62052

Subjects
Primary: 62G07: Density estimation
Secondary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

Keywords
Regression splines least squares Bernoulli polynomial asymptotic bias asymptotic variance asymptotic normality confidence interval and confidence band

Citation

Zhou, S.; Shen, X.; Wolfe, D.A. Local Asymptotics for Regression Splines and Confidence Regions. Ann. Statist. 26 (1998), no. 5, 1760--1782. doi:10.1214/aos/1024691356. https://projecteuclid.org/euclid.aos/1024691356


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References

  • AGARWAL, G. G. and STUDDEN W. J. 1980. Asy mptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist. 8 1307 1325. Z.
    Mathematical Reviews (MathSciNet): MR594647
    Zentralblatt MATH: 0522.62032
    Project Euclid: euclid.aos/1176345203
  • BARROW, D. L. and SMITH, P. W. 1978. Asy mptotic properties of best L 0, 1 approximation by 2 spline with variable knots. Quart. Appl. Math. 36 293 304. Z.
    Mathematical Reviews (MathSciNet): MR508773
    Zentralblatt MATH: 0395.41012
  • CRAVEN, P. and WAHBA, G. 1979. Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the generalized cross-validation. Numer. Math. 31 337 403. Z.
    Mathematical Reviews (MathSciNet): MR516581
    Zentralblatt MATH: 0377.65007
  • DE BOOR, C. 1972. On calculating with B-splines. J. Approx. Theory 6 50 62. Z.
    Mathematical Reviews (MathSciNet): MR49:3381
    Zentralblatt MATH: 0239.41006
  • DE BOOR, C. 1978. A Practical Guide to Splines. Springer, New York. Z.
    Mathematical Reviews (MathSciNet): MR80a:65027
  • DEMKO, S. 1977. Inverses of band matrices and local convergence of spline projection. SIAM J. Numer. Anal. 14 616 619. Z. Z.
    Mathematical Reviews (MathSciNet): MR56:13520
    Zentralblatt MATH: 0367.65024
    JSTOR: links.jstor.org
  • FRIEDMAN, J. H. 1991. Multivariate adaptive regression splines with discussion. Ann. Statist. 19 1 141. Z.
    Mathematical Reviews (MathSciNet): MR1091842
    Zentralblatt MATH: 0765.62064
    Project Euclid: euclid.aos/1176347963
  • GAENSSLER, P. and WELLNER, J. A. 1981. Glivenko Cantelli theorems. Ency clopedia of Statistical Science 3 442 445. Wiley, New York. Z.
  • GASSER, T. and MULLER, H. G. 1984. Estimating regression functions and their derivatives by ¨ the kernel method. Scand. J. Statist. 11 171 185.
    Mathematical Reviews (MathSciNet): MR86h:62056
  • GASSER, T., SROKA, L. and JENNEN-STEINMETZ, C. 1986. Residual variance and residual pattern in nonlinear regression. Biometrika 73 625 633. Z.
    Mathematical Reviews (MathSciNet): MR88k:62107
    Zentralblatt MATH: 0649.62035
    JSTOR: links.jstor.org
  • HALL, P., KAY, J. W. and TITTERINGTON, D. M. 1990. Asy mptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521 528. Z.
    Mathematical Reviews (MathSciNet): MR1087842
    JSTOR: links.jstor.org
  • HUANG, S. Y. and STUDDEN, W. Y. 1993. An equivalent kernel method for least squares spline Z. regression. Statist. Decisions 3 179 201 Supplement. Z.
    Mathematical Reviews (MathSciNet): MR1244071
  • JOHANSEN, S. and JOHNSTONE, M. 1990. Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652 684. Z.
    Mathematical Reviews (MathSciNet): MR92d:62018
    Zentralblatt MATH: 0723.62018
    Project Euclid: euclid.aos/1176347620
  • KOENKER, R., NG, P. and PORTNOY, S. 1994. Non-parametric estimation of conditional quantile functions. Biometrika 81 673 680. Z.
    Mathematical Reviews (MathSciNet): MR1326417
    Zentralblatt MATH: 0810.62040
    JSTOR: links.jstor.org
  • KOOPERBERG, C., STONE, C. and TRUONG, Y. K. 1995. The L rate of convergence for hazard 2 regression. Scand. J. Statist. 22 143 157. Z.
    Zentralblatt MATH: 0839.62050
    Mathematical Reviews (MathSciNet): MR1339748
  • NADARAy A, E. A. 1964. On estimating regression. Theory Probab. Appl. 9 141 142. Z.
  • PORTNOY, S. 1997. Local asy mptotics for quantile smoothing splines. Ann. Statist. 25 414 434. Z.
    Mathematical Reviews (MathSciNet): MR1429932
    Zentralblatt MATH: 0898.62044
    Project Euclid: euclid.aos/1034276636
  • SCHUMAKER, L. L. 1981. Spline Functions: Basic Theory. Wiley, New York. Z.
    Mathematical Reviews (MathSciNet): MR82j:41001
  • SHEN, X. and HU, C. 1995. Universal sieves and spline adaptation. Technical Report 543, Dept. Statist., Ohio State Univ. Z.
  • SHI, P. and LI, G. 1994. On the rate of convergence of ``minimum L -norm'' estimates in a 1 partly linear model. Comm. Statist. Theory Methods 23 175 196. Z.
    Mathematical Reviews (MathSciNet): MR1261740
    Zentralblatt MATH: 0825.62077
  • SHI, P. and LI, G. 1995. Global convergence rates of B-spline M-estimators in nonparametric regression. Statist. Sinica 5 303 318. Z.
    Mathematical Reviews (MathSciNet): MR96e:62067
    Zentralblatt MATH: 0824.62035
  • STONE, C. 1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040 1053. Z.
    Mathematical Reviews (MathSciNet): MR673642
    Zentralblatt MATH: 0511.62048
    Project Euclid: euclid.aos/1176345969
  • STONE, C. 1985. Additive regression and other nonparametric models. Ann. Statist. 13 689 705. Z.
    Mathematical Reviews (MathSciNet): MR87i:62111
    Zentralblatt MATH: 0605.62065
    Project Euclid: euclid.aos/1176349548
  • STONE, C. 1986. The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590 606. Z.
    Mathematical Reviews (MathSciNet): MR88a:62131
    Zentralblatt MATH: 0603.62050
    Project Euclid: euclid.aos/1176349940
  • STONE, C. 1994. The use of poly nomial splines and their tensor products in multivariate function estimation. Ann. Statist. 22 118 184. Z.
    Mathematical Reviews (MathSciNet): MR1272079
    Zentralblatt MATH: 0827.62038
    Project Euclid: euclid.aos/1176325361
  • STONE, C. and KOO, C. Y. 1986. Additive splines in statistics. In Proceedings of the Statistical Computing Section 45 48. Amer. Statist. Assoc., Washington, DC.
  • COLUMBUS, OHIO 43210-1247 OHIO STATE UNIVERSITY E-MAIL: zhou@stat.ohio-state.edu COLUMBUS, OHIO 43210-1247

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