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Modelling time trend via spline confidence band

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Abstract

Simultaneous confidence band is obtained for the trend function of time series with heteroscedastic α-mixing errors, based on constant and linear spline smoothing. Simulation study confirms that the bands have conservative coverage of the true trend function. Linear band has been constructed for the leaf area index (LAI) data collected in East Africa, which has revealed that the trigonometric curve in the regional atmospheric modelling system (RAMS) is inadequate.

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References

  • Beran J., Feng Y. (2002) Local polynomial fitting with long-memory, short-memory and antipersistent errors. The Annals of the Institute of Statistical Mathematics 54: 291–311

    Article  MathSciNet  MATH  Google Scholar 

  • Beran J., Feng Y. (2002) SEMIFAR models—A semiparametric framework for modelling trends, long-range dependence and nonstationarity. Computational Statistics and Data Analysis 40: 393–419

    Article  MathSciNet  MATH  Google Scholar 

  • Bickel P.J., Rosenblatt M. (1973) On some global measures of the deviations of density function estimates. Annals of Statistics 1: 1071–1095

    Article  MathSciNet  MATH  Google Scholar 

  • Bosq D. (1996) Nonparametric statistics for stochastic processes. Springer, New York

    MATH  Google Scholar 

  • Cai Z. (2002) Regression quantiles for time series. Econometric Theory 18: 169–192

    Article  MathSciNet  MATH  Google Scholar 

  • Claeskens G., Van Keilegom I. (2003) Bootstrap confidence bands for regression curves and their derivatives. Annals of Statistics 31: 1852–1884

    Article  MathSciNet  MATH  Google Scholar 

  • de Boor C. (2001) A practical guide to splines. Springer, New York

    MATH  Google Scholar 

  • Diack C. (2001) Testing the shape of a regression curve. Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 333(7): 677–680

    Article  MathSciNet  MATH  Google Scholar 

  • Fan J., Gijbels I. (1996) Local polynomial modelling and its applications. Chapman and Hall, London

    MATH  Google Scholar 

  • Fan J., Yao Q. (2003) Nonlinear time series. Springer, New York

    MATH  Google Scholar 

  • Feng Y. (2004) Simultaneously modeling conditional heteroskedasticity and scale change. Econometric Theory 20: 563–596

    Article  MathSciNet  MATH  Google Scholar 

  • Gantmacher F.R., Krein M.G. (1960) Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme. Akademie, Berlin

    Google Scholar 

  • Härdle W. (1989) Asymptotic maximal deviation of M-smoothers. Journal of Multivariate Analysis 29: 163–179

    Article  MathSciNet  MATH  Google Scholar 

  • Härdle W. (1990) Applied nonparametric regression. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Härdle W., Marron J.S., Yang L. (1997) Discussion of “Polynomial splines and their tensor products in extended linear modeling” by Stone et al. The Annals of Statistics 25: 1443–1450

    Google Scholar 

  • Härdle W., Huet S., Mammen E., Sperlich S. (2004) Bootstrap inference in semiparametric generalized additive models. Econometric Theory 20: 265–300

    Article  MathSciNet  MATH  Google Scholar 

  • Huang J.Z. (1998) Projection estimation in multiple regression with application to functional ANOVA models. Annals of Statistics 26: 242–272

    Article  MathSciNet  MATH  Google Scholar 

  • Huang J.Z. (2003) Local asymptotics for polynomial spline regression. Annals of Statistics 31: 1600–1635

    Article  MathSciNet  MATH  Google Scholar 

  • Huang J.Z., Yang L. (2004) Identification of nonlinear additive autoregressive models. Journal of the Royal Statistical Society Series B 66: 463–477

    Article  MathSciNet  MATH  Google Scholar 

  • Huber-Carol C., Balakrishnan N., Nikulin M., Mesbah M. (2002) Goodness-of-fit tests and model validity. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  • Johnson R.A., Wichern D.W. (1992) Applied multivariate statistical analysis. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  • Leadbetter M.R., Lindgren G., Rootzén H. (1983) Extremes and related properties of random sequences and processes. Springer, New York

    Book  MATH  Google Scholar 

  • Liang H., Uña-Álvarez J. (2009) A Berry–Esseen type bound in kernel density estimation for strong mixing censored samples. Journal of Multivariate Analysis 100: 1219–1231

    Article  MathSciNet  MATH  Google Scholar 

  • Liebscher E. (1999) Asymptotic normality of nonparametric estimators under mixing condition. Statistics & Probability Letters 43: 243–250

    Article  MathSciNet  MATH  Google Scholar 

  • Liebscher E. (2001) Estimation of the density and the regression function under mixing conditions. Statistics & Decisions 19: 9–26

    MathSciNet  MATH  Google Scholar 

  • Masry E., Fan J. (1997) Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics 24: 165–179

    Article  MathSciNet  MATH  Google Scholar 

  • Olson J.M., Alagarswamy G., Andresen J.A., Campbell D.J., Davis A.Y., Ge J. et al (2008) Integrating diverse methods to understand climate–land interactions in East Africa. Geoforum 39: 898–911

    Article  Google Scholar 

  • Paparoditis E., Politis D. (2000) The local bootstrap for kernel estimators under general dependence conditions. The Annals of Institute of Statistical Mathematics 52: 139–259

    Article  MathSciNet  MATH  Google Scholar 

  • Roussas G.G. (1988) Nonparametric estimation in mixing sequences of random Variables. Journal of Statistical Planning and Inference 18: 135–149

    Article  MathSciNet  MATH  Google Scholar 

  • Roussas G.G. (1990) Nonparametric regression estimation under mixing conditions. Stochastic Processes and Their Applications 36: 107–116

    Article  MathSciNet  MATH  Google Scholar 

  • Schumway R., Stoffer D. (2006) Time series analysis and its applications. Springer, New York

    Google Scholar 

  • Silverman B.W. (1986) Density estimation for statistics and data analysis. Chapman and Hall, London

    MATH  Google Scholar 

  • Song Q., Yang L. (2009) Spline confidence bands for variance function. Journal of Nonparametric Statistics 21: 589–609

    Article  MathSciNet  MATH  Google Scholar 

  • Stone C.J. (1985) Additive regression and other nonparametric models. Annals of Statistics 13: 689–705

    Article  MathSciNet  MATH  Google Scholar 

  • Stone C.J. (1994) The use of polynomial splines and their tensor products in multivariate function estimation. Annals of Statistics 22: 118–184

    Article  MathSciNet  MATH  Google Scholar 

  • Sunklodas J. (1984) On the rate of convergence in the central limit theorem for strongly mixing random variables. Lithuanian Mathematical Journal 24: 182–190

    Article  MathSciNet  MATH  Google Scholar 

  • Tusnády G. (1977) A remark on the approximation of the sample df in the multidimensional case. Periodica Mathematica Hungarica 8: 53–55

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, J. (2009). Modelling time trend via spline confidence band. Manuscript, 26 pages. http://www.math.uic.edu/~wangjing/bandfixedfull.pdf.

  • Wang J., Yang L. (2009) Polynomial spline confidence bands for regression curves. Statistica Sinica 19: 325–342

    MathSciNet  MATH  Google Scholar 

  • Wang J., Yang L. (2009) Efficient and fast spline-backfitted kernel smoothing of additive models. Annals of the Institute of Statistical Mathematics 61: 663–690

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, J., Qi, J., Yang, L., Olson, J., Nathan, M., Nathan, T., et al. (2006). Derivation of phenological information from remotely sensed imagery for improved regional climate modeling. Manuscript.

  • Xia Y. (1998) Bias-corrected confidence bands in nonparametric regression. Journal of the Royal Statistical Society: Series B 60: 797–811

    Article  MATH  Google Scholar 

  • Xue L., Yang L. (2006) Additive coefficient modeling via polynomial spline. Statistica Sinica 16: 1423–1446

    MathSciNet  Google Scholar 

  • Yang L. (2008) Confidence band for additive regression model. Journal of Data Science 6: 207–217

    Google Scholar 

  • Zhang F. (1999) Matrix theory: Basic results and techniques. Springer, New York

    MATH  Google Scholar 

  • Zhou S., Shen X., Wolfe D.A. (1998) Local asymptotics of regression splines and confidence regions. The Annals of Statistics 26: 1760–1782

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Jing Wang

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  1. Jing Wang
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Correspondence to Jing Wang.

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Wang, J. Modelling time trend via spline confidence band. Ann Inst Stat Math 64, 275–301 (2012). https://doi.org/10.1007/s10463-010-0311-8

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  • DOI: https://doi.org/10.1007/s10463-010-0311-8

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