TEST

, Volume 23, Issue 4, pp 806–843 | Cite as

A simultaneous confidence corridor for varying coefficient regression with sparse functional data

  • Lijie Gu
  • Li Wang
  • Wolfgang K. Härdle
  • Lijian Yang
Original Paper

Abstract

We consider a varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time-independent covariates with coefficients as functions of time-dependent covariates. Based on spline smoothing, we propose data-driven simultaneous confidence corridors for the coefficient functions with asymptotically correct confidence level. Such confidence corridors are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypotheses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p values on the effects of smoking, pre-infection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment.

Keywords

B spline Confidence corridor Karhunen–Loève \(L^{2}\) representation Knots Functional data Varying coefficient 

Mathematics Subject Classification (2000)

62G08 62G15 62G32 

References

  1. Bosq D (1998) Nonparametric statistics for stochastic processes. Springer, New YorkCrossRefMATHGoogle Scholar
  2. Brumback B, Rice JA (1998) Smoothing spline models for the analysis of nested and crossed samples of curves (with Discussion). J Am Stat Assoc 93:961–994CrossRefMATHMathSciNetGoogle Scholar
  3. Cao G, Yang L, Todem D (2012) Simultaneous inference for the mean function based on dense functional data. J Nonparametr Stat 24:359–377CrossRefMATHMathSciNetGoogle Scholar
  4. Cao G, Wang J, Wang L, Todem D (2012) Spline confidence bands for functional derivatives. J Stat Plan Inference 142:1557–1570CrossRefMATHMathSciNetGoogle Scholar
  5. Chiang CT, Rice JA, Wu CO (2001) Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables. J Am Stat Assoc 96:605–619CrossRefMATHMathSciNetGoogle Scholar
  6. Claeskens G, Van Keilegom I (2003) Bootstrap confidence bands for regression curves and their derivatives. Ann Stat 31:1852–1884CrossRefMATHGoogle Scholar
  7. de Boor C (2001) A practical guide to splines. Springer, New YorkMATHGoogle Scholar
  8. Fan J, Zhang JT (2000) Functional linear models for longitudinal data. J R Stat Soc Ser B 62:303–322CrossRefMathSciNetGoogle Scholar
  9. Fan J, Zhang WY (2000) Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand J Stat 27:715–731CrossRefMATHMathSciNetGoogle Scholar
  10. Fan J, Zhang WY (2008) Statistical methods with varying coefficient models. Stat Interface 1:179–195CrossRefMathSciNetGoogle Scholar
  11. Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, New YorkGoogle Scholar
  12. Gabrys R, Horváth L, Kokoszka P (2010) Tests for error correlation in the functional linear model. J Am Stat Assoc 105:1113–1125CrossRefGoogle Scholar
  13. Hall P, Müller HG, Wang JL (2006) Properties of principal component methods for functional and longitudinal data analysis. Ann Stat 34:1493–1517CrossRefMATHGoogle Scholar
  14. Hall P, Titterington DM (1988) On confidence bands in nonparametric density estimation and regression. J Mult Anal 27:228–254CrossRefMATHMathSciNetGoogle Scholar
  15. Härdle W, Luckhaus S (1984) Uniform consistency of a class of regression function estimators. Ann Stat 12:612–623CrossRefMATHGoogle Scholar
  16. Hastie T, Tibshirani R (1993) Varying-coefficient models. J R Stat Soc Ser B 55:757–796MATHMathSciNetGoogle Scholar
  17. Hoover DR, Rice JA, Wu CO, Yang LP (1998) Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85:809–822CrossRefMATHMathSciNetGoogle Scholar
  18. Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, New YorkCrossRefMATHGoogle Scholar
  19. Huang JZ, Wu CO, Zhou L (2002) Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89:111–128CrossRefMATHMathSciNetGoogle Scholar
  20. Huang JZ, Wu CO, Zhou L (2004) Polynomial spline estimation and inference for varying coefficient models with longitudinal data. Stat Sin 14:763–788MATHMathSciNetGoogle Scholar
  21. James GM, Hastie T, Sugar C (2000) Principal component models for sparse functional data. Biometrika 87:587–602CrossRefMATHMathSciNetGoogle Scholar
  22. James GM, Sugar CA (2003) Clustering for sparsely sampled functional data. J Am Stat Assoc 98:397–408CrossRefMATHMathSciNetGoogle Scholar
  23. Leadbetter MR, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, New YorkCrossRefMATHGoogle Scholar
  24. Liu R, Yang L (2010) Spline-backfitted kernel smoothing of additive coefficient model. Econ Theory 26:29–59CrossRefMATHGoogle Scholar
  25. Ma S, Yang L, Carroll RJ (2012) A simultaneous confidence band for sparse longitudinal regression. Stat Sin 22:95–122MATHMathSciNetGoogle Scholar
  26. Manteiga W, Vieu P (2007) Statistics for functional data. Comput Stat Data Anal 51:4788–4792CrossRefMATHMathSciNetGoogle Scholar
  27. Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, New YorkGoogle Scholar
  28. Wang L, Li H, Huang JZ (2008) Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. J Am Stat Assoc 103:1556–1569CrossRefMATHMathSciNetGoogle Scholar
  29. Wang L, Yang L (2009) Polynomial spline confidence bands for regression curves. Stat Sin 19:325–342MATHGoogle Scholar
  30. Wu CO, Chiang CT (2000) Kernel smoothing on varying coefficient models with longitudinal dependent variable. Stat Sin 10:433–456MATHMathSciNetGoogle Scholar
  31. Wu CO, Chiang CT, Hoover DR (1998) Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. J Am Stat Assoc 93:1388–1402CrossRefMATHMathSciNetGoogle Scholar
  32. Wu Y, Fan J, Müller HG (2010) Varying-coefficient functional linear regression. Bernoulli 16:730–758CrossRefMATHMathSciNetGoogle Scholar
  33. Xue L, Yang L (2006) Additive coefficient modelling via polynomial spline. Stat Sin 16:1423–1446MathSciNetGoogle Scholar
  34. Xue L, Zhu L (2007) Empirical likelihood for a varying coefficient model with longitudinal data. J Am Stat Assoc 102:642–654CrossRefMATHMathSciNetGoogle Scholar
  35. Yao W, Li R (2013) New local estimation procedure for a non-parametric regression function for longitudinal data. J R Stat Soc Ser B 75:123–138CrossRefMathSciNetGoogle Scholar
  36. Yao F, Müller HG, Wang JL (2005a) Functional linear regression analysis for longitudinal data. Ann Stat 33:2873–2903CrossRefMATHGoogle Scholar
  37. Yao F, Müller HG, Wang JL (2005b) Functional data analysis for sparse longitudinal data. J Am Stat Assoc 100:577–590CrossRefMATHGoogle Scholar
  38. Zhou L, Huang J, Carroll RJ (2008) Joint modelling of paired sparse functional data using principal components. Biometrika 95:601–619CrossRefMATHMathSciNetGoogle Scholar
  39. Zhu H, Li R, Kong L (2012) Multivariate varying coefficient model for functional responses. Ann Stat 40:2634–2666CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  • Lijie Gu
    • 1
  • Li Wang
    • 2
  • Wolfgang K. Härdle
    • 3
    • 4
  • Lijian Yang
    • 1
  1. 1.Center for Advanced Statistics and Econometrics ResearchSoochow UniversitySuzhouChina
  2. 2.Department of StatisticsIowa State UniversityAmesUSA
  3. 3.Center for Applied Statistics and Economics (C.A.S.E.)Humboldt-Universität zu BerlinBerlinGermany
  4. 4.Lee Kong Chian School of BusinessSingapore Management UniversitySingaporeSingapore

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