A smooth simultaneous confidence corridor for the mean of sparse functional data

Journal of the American Statistical Association

Volumn 109, Issue 506, 2014, Pages 661-673

  Zheng, Shuzhuan ^a,b   Yang, Lijian ^a,b   Härdle, Wolfgang K ^c,d  

^a SOOCHOW UNIVERSITY   (China)

^b MICHIGAN STATE UNIVERSITY   (United States)

^c HUMBOLDT UNIVERSITY   (Germany)

^d SINGAPORE MANAGEMENT UNIVERSITY   (Singapore)

Author keywords

Double sum; Extreme value; Karhunen Lo ve L2 representation; Local linear estimator; Strong approximation

Indexed keywords

EID: 84987916084     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.2013.866899     Document Type: Article

References (39)

1. Azäıs, J. M., and Wschebor, M. (2009), Level Sets and Extrema of Rand om Processes and Fields, Hoboken, NJ: Wiley.
2. Bachrach, L., Hastie, T., Wang, W., Narasimhan, B., and Marcus, R. (1999), “Bone Mineral Acquisition in Healthy Asian, Hispanic, Black, and Caucasian Youth: A Longitudinal Study,” Clinical Endocrinology & Metabolism, 84, 4702–4712.
3. Bickel, P. J., and Rosenblatt, M. (1973), “On Some Global Measures of the Deviations of Density Function Estimates,” The Annals of Statistics, 1, 1071–1095.
4. Bosq, D. (1996), Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, New York: Springer-Verlag.
5. Cai, T., and Hall, P. (2006), “Prediction in Functional Linear Regression,” The Annals of Statistics, 34, 2159–2179.
6. Cao, G., Wang, J., Wang, L., and Todem, D. (2012), “Spline Confidence Band s for Functional Derivatives,” Journal of Statistical Planning and Inference, 142, 1557–1570.
7. Cao, G., Yang, L., and Todem, D. (2012), “Simultaneous Inference for the Mean FunctionBased on Dense FunctionalData,” Journal of Nonparametric Statistics, 24, 359–377.
8. Cardot, H., Ferraty, F., and Sarda, P. (2003), “Spline Estimators for the Functional Linear Model,” Statistica Sinica, 13, 571–591.
9. Cierco-Ayrolles, C., Croquette, A., and Delmas, C. (2003), “Computing the Distribution of the Maximum ofGaussian Rand om Processes,” Methodology and Computing in Applied Probability, 5, 427–438.
10. Claeskens, G., and Van Keilegom, I. (2003), “Bootstrap Confidence Band s for Regression Curves and Their Derivatives,” The Annals of Statistics, 31, 1852–1884.
11. Csőrgő, M., and Révész, P. (1981), Strong Approximations in Probability and Statistics, New York-London: Academic Press.
12. Fan, J., and Gijbels, I. (1996), Local PolynomialModelling and Its Applications, London: Chapman and Hall.
13. Fan, J., and Yao, Q. (2005), Nonlinear Time Series, New York: Springer-Verlag.
14. Fan, J., and Zhang, W. Y. (2000), “Simultaneous Confidence Band s and Hypothesis Testing in Varying-Coefficient Models,” Scand inavian Journal of Statistics, 27, 715–731.
15. Ferraty, F., and Vieu, P. (2006), Nonparametric Functional Data Analysis: Theory and Practice, Berlin: Springer.
16. Hall, P. (1993), “On Edgeworth Expansion and Bootstrap Confidence Band s in Nonparametric Regression,” Journal of the Royal Statistical Society, Series B, 55, 291–304.
17. James, G. M., Hastie, T., and Sugar, C. (2000), “Principal Component Models for Sparse Functional Data,” Biometrika, 87, 587–602.
18. Johnston, G. J. (1982), “Probabilities of Maximal Deviations for Nonparametric Regression Function Estimates,” Journal of Multivariate Analysis, 12, 402–414.
19. Kerkyacharian, G., Nickl, R., and Picard, D. (2012), “Concentration Inequalities and Confidence Band s for Needlet Density Estimators on Compact Homogeneous Manifolds,” Probability Theory and Related Fields, 153, 363–404.
20. Kreiss, J. P., and Neumann, M. H. (1998), “Regression-Type Inference in Nonparametric Autoregression,” The Annals of Statistics, 26, 1570–1613.
21. Li, Y., and Hsing, T. (2010), “Uniform Convergence Rates for Nonparametric Regression and Principal Component Analysis in Functional/Longitudinal Data,” The Annals of Statistics, 38, 3321–3351.
22. Lucas, P.W., and Diggle, P. J. (1997), “TheUse of LongitudinalData Analysis to Study theMulti-seasonal Growth Responses of Norway and Sitka Spruce to Summer Exposure to Ozone: Implications for the Determination of Critical Levels,” New Phytollogist, 137, 315–323.
23. Ma, S., Yang, L., and Carroll, R. J. (2012), “Simultaneous Confidence Band for Sparse Longitudinal Regression,” Statistica Sinica, 22, 95–122.
24. Müller, H. G. (2009), “Functional Modeling of Longitudinal Data,” in Longitudinal Data Analysis, Hand books of Modern Statistical Methods, NewYork: Wiley, pp. 223–252.
25. Neumann, M. H., and Polzehl, J. (1998), “Simultaneous Bootstrap Confidence Band s in Nonparametric Regression,” Journal of Nonparametric Statistics, 9, 307–333.
26. Paul, D., and Peng, J. (2009), “Consistency of Restricted Maximum Likelihood Estimators of Principale Components,” The Annals of Statistics, 37, 1229–1271.
27. Piterbarg, V. I. (1996), Asymptotic Methods in the Theory of Gaussian Processes and Fields, Providence, RI: American Mathematical Society.
28. Sluis, I., Ridder, M., Boot, A., Krenning, P., and Keizer-Schrama, S. (2002), “Reference Data for Bone Density and Body Composition Measured With Dual Energy X Ray Absorptiometry in White Children and Young Adults,” Archives of Disease in Childhood, 87, 341–347.
29. Song, Q., and Yang, L. (2009), “Spline Confidence Band s for Variance Function,” Journal of Nonparametric Statistics, 21, 589–609.
30. Wang, J., and Yang, L. (2009), “Polynomial Spline Confidence Band s for Regression Curves,” Statistica Sinica, 19, 325–342.
31. Wang, J., and Yang, L. (2007), “Spline-Backfitted Kernel Smoothing of Nonlinear Additive Autoregression Model,” The Annals of Statistics, 35, 2474–2503.
32. Wang, N., Carroll, R. J., and Lin, X. (2005), “Efficient Semiparametric Marginal Estimation for Longitudinal/Clustered Data,” Journal of the American Statistical Association, 100, 147–157.
33. Wu, W., and Zhao, Z. (2007), “Inference of Trends in Time Series,” Journal of the Royal Statistical Society, Series B, 69, 391–410.
34. Xia, Y. (1998), “Bias-Corrected Confidence Band s in Nonparametric Regression,” Journal of the Royal Statistical Society, Series B, 60, 797–811.
35. Xue, L., and Yang, L. (2006), “Estimation of Semiparametric Additive Coefficient Model,” Journal of Statistical Planning and Inference, 136, 2506–2534.
36. Yao, F. (2007), “Asymptotic Distributions of Nonparametric Regression Estimators for Longitudinal or Functional Data,” Journal of Multivariate Analysis, 98, 40–56.
37. Yao, F., Müller, H. G., and Wang, J. L. (2005a), “Functional Linear Regression Analysis for Longitudinal Data,” The Annals of Statistics, 33, 2873–2903.
38. Yao, F., Müller, H. G., and Wang, J. L (2005b), “Functional Data Analysis for Sparse Longitudinal Data,” Journal of the American Staistical Association, 100, 577–590.
39. Zhao, Z., and Wu, W. (2008), “Confidence Band s in Nonparametric Time Series Regression,” The Annals of Statistics, 36, 1854–1878.