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Consistency of cross-validation when the data are curves

Author

Listed:
  • Hart, Jeffrey D.
  • Wehrly, Thomas E.

Abstract

Suppose one observes a random sample of n continuous time Gaussian processes on the interval [0, 1]; in other words, each observation is a curve. Of interest is estimating the common mean function of the processes by a kernel smoother. The bandwidth of the kernel estimator is chosen by a version of cross-validation in which deleting an observation means deleting one of the n curves. It is shown that using this form of cross-validation leads to an asymptotically optimal choice of bandwidth. This result is contrasted with the inconsistency of cross-validation in a seemingly more tractable problem.

Suggested Citation

  • Hart, Jeffrey D. & Wehrly, Thomas E., 1993. "Consistency of cross-validation when the data are curves," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 351-361, April.
    • Handle: RePEc:eee:spapps:v:45:y:1993:i:2:p:351-361
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    Citations

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    Cited by:

    1. Ferreira, Eva & Núñez-Antón, Vicente & Rodríguez-Póo, Juan, 1997. "Kernel regression estimates of growth curves using nonstationary correlated errors," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 413-423, June.
    2. Jianqing Fan & Qiwei Yao & Zongwu Cai, 2003. "Adaptive varying‐coefficient linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 57-80, February.
    3. Rui Li & Yuanyuan Zhang, 2021. "Two-stage estimation and simultaneous confidence band in partially nonlinear additive model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1109-1140, November.
    4. Colin Wu & Xin Tian & Jarvis Yu, 2010. "Nonparametric estimation for time-varying transformation models with longitudinal data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(2), pages 133-147.
    5. Vicente Núñez-Antón & Juan Rodríguez-Póo & Philippe Vieu, 1999. "Longitudinal data with nonstationary errors: a nonparametric three-stage approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 201-231, June.
    6. Colin O. Wu & Kai F. Yu, 2002. "Nonparametric Varying-Coefficient Models for the Analysis of Longitudinal Data," International Statistical Review, International Statistical Institute, vol. 70(3), pages 373-393, December.
    7. Ferreira García, María Eva & Núñez Antón, Vicente Alfredo & Rodríguez Poo, Juan M., 1999. "Two-Stage Nonparametric Regression for Longitudinal Data," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    8. Cardot, Hervé & De Moliner, Anne & Goga, Camelia, 2015. "Estimating with kernel smoothers the mean of functional data in a finite population setting. A note on variance estimation in presence of partially observed trajectories," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 156-166.
    9. Gerard, Patrick D. & Schucany, William R., 1996. "On combining independent nonparametric regression estimators," Statistics & Probability Letters, Elsevier, vol. 26(1), pages 25-34, January.

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