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Testing for the Mean of Random Curves: A Penalization Approach

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Abstract

Let X 1,...,X n be an i.i.d. sample of random curves, viewed as Hilbert space valued random elements, with mean curve m. An asymptotic test of m = m 0 vs m ≠  m 0 is proposed, when m 0 is a fixed known function. The test statistics converges under very mild assumptions and relies on the pseudo-inversion of the covariance operator (leading to a non standard inverse problem). The power against local alternatives is investigated.

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References

  • Araujo, A. and Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley and Sons, 1980.

  • V. Arsenin A. Tikhonov (1977) Solutions of Ill-posed Problems Winston and Sons Washington, D.C Occurrence Handle0354.65028

    MATH  Google Scholar 

  • P. Besse H. Cardot D.B. Stephenson (2000) ArticleTitleAutoregressive forecasting of some functional climatic variations Scand. Journal Stat. 27 673–687 Occurrence Handle0962.62089 Occurrence Handle10.1111/1467-9469.00215

    Article  MATH  Google Scholar 

  • Billingsley, P.: Convergence of Probability Measures, Wiley Series in Probability and Mathematical Statistics, 1968.

  • Bosq, D.: Modelization, non-parametric estimation and prediction for continuous time processes. In: Roussas (ed.), Nonparametric Functional Estimation and Related Topics, NATO, ASI Series, 1991, pp. 509–529.

  • Bosq, D.: Linear Processes in Function Spaces, Lecture Notes in Statistics, Springer-Verlag, 2000.

  • H. Cardot F. Ferraty P. Sarda (1999) ArticleTitleFunctional linear model Stat. Probab. Lett. 45 11–22 Occurrence Handle0962.62081 Occurrence Handle1718346 Occurrence Handle10.1016/S0167-7152(99)00036-X

    Article  MATH  MathSciNet  Google Scholar 

  • H. Cardot F. Ferraty A. Mas P. Sarda (2002) ArticleTitleTesting hypotheses in the functional linear model Scand. J. Stat. 99 1–15

    Google Scholar 

  • Cavallini, A., Montanari, G. C., Loggini, M., Lessi, O. and Cacciari, M.: Nonparametric prediction of harmonic levels in electrical networks. In: Proceedings of the 6th IEEE ICHPS, Bologna, 1994.

  • J. Dauxois A. Pousse Y. Romain (1982) ArticleTitleAsymptotic theory for the principal component analysis of a random vector function: some applications to statistical inference J. Multivariate Anal. 12 136–154 Occurrence Handle0539.62064 Occurrence Handle650934 Occurrence Handle10.1016/0047-259X(82)90088-4

    Article  MATH  MathSciNet  Google Scholar 

  • R.M. Dudley (1968) ArticleTitleDistances of probability measures and random variables Ann. Math. Stat. 39 1563–1572 Occurrence Handle0169.20602 Occurrence Handle230338

    MATH  MathSciNet  Google Scholar 

  • N. Dunford J.T. Schwartz (1962) Linear operators NumberInSeriesVol 2 Wiley NewYork

    Google Scholar 

  • J. Fan (1996) ArticleTitleTest of significance based on Wavelet thresholding and Neyman’s truncation JASA 91 674–688 Occurrence Handle0869.62032

    MATH  Google Scholar 

  • J. Fan S.K. Lin (1998) ArticleTitleTest of significance when data are curves JASA 93 1007–1022 Occurrence Handle1064.62525 Occurrence Handle1649196

    MATH  MathSciNet  Google Scholar 

  • I.E. Franck J.H. Friedman (1993) ArticleTitleA statistical view of some chemometrics regression tools Technometrics 35 109–148 Occurrence Handle10.2307/1269656

    Article  Google Scholar 

  • C. Groetsch (1993) Inverse Problems in the Mathematical Sciences Vieweg Wiesbaden Occurrence Handle0779.45001

    MATH  Google Scholar 

  • A. Kneip T. Gasser (1992) ArticleTitleStatistical tools to analyze data representing a sample of curves Ann. Stat. 20 1266–1305 Occurrence Handle0785.62042 Occurrence Handle1186250

    MATH  MathSciNet  Google Scholar 

  • O.V. Lepski A.B. Tsybakov (2000) ArticleTitleAsymptotically exact nonparametric hypothesis tesing in sup-norm and at a fixed point Probab. Theory Rel. 117 17–48 Occurrence Handle0971.62022 Occurrence Handle1759508 Occurrence Handle10.1007/s004400050265

    Article  MATH  MathSciNet  Google Scholar 

  • A. Mas (2002a) ArticleTitleRates of weak convergence for images of measures by families of mappings Stat. Probabil. Lett. 56 7–12 Occurrence Handle1881125 Occurrence Handle10.1016/S0167-7152(01)00038-4

    Article  MathSciNet  Google Scholar 

  • A. Mas (2002b) ArticleTitleWeak convergence for the covariance operators of a Hilbertian linear process Stoch. Proc. and Appl. 99 117–135 Occurrence Handle1894254 Occurrence Handle10.1016/S0304-4149(02)00087-X

    Article  MathSciNet  Google Scholar 

  • M.Z. Nashed G. Wahba (1974) ArticleTitleGeneralized inverses in reproducing kernel spaces: An approach to regularization of linear operator equations SIAM J. Math. Anal 6 974–987 Occurrence Handle358405 Occurrence Handle10.1137/0505095

    Article  MathSciNet  Google Scholar 

  • J.O. Ramsay B.W. Silverman (1997) Functional Data Analysis Springer-Verlag Berlin

    Google Scholar 

  • J.A. Rice B.W. Silverman (1991) ArticleTitleEstimating the mean and covariance structure nonparametrically when the data are curves J. Roy. Stat. Soc. Ser. B 53 233–243 Occurrence Handle0800.62214 Occurrence Handle1094283

    MATH  MathSciNet  Google Scholar 

  • W. Rudin (1973) Functional Analysis Mc Graw-Hill New-York Occurrence Handle0867.46001

    MATH  Google Scholar 

  • B.W. Silverman (1996) ArticleTitleSmoothed principal component Analysis by choice of norm Ann. Stat. 24 1–24 Occurrence Handle0853.62044 Occurrence Handle10.1214/aos/1033066196

    Article  MATH  Google Scholar 

  • V.G. Spokoiny (1996) ArticleTitleAdaptive hypothesis testing using wavelets Ann. Stat 24 2477–2498 Occurrence Handle0898.62056 Occurrence Handle1425962 Occurrence Handle10.1214/aos/1032181163

    Article  MATH  MathSciNet  Google Scholar 

  • Vakhania, N. N., Tarieladze, V. I. and Chobanyan, S. M.: Probability Distributions on Banach Spaces , Mathematics and its Applications, D. Reidel Publishing, 1987.

  • W. Whitt (1974) ArticleTitlePreservation of rates of convergence under mappings Z. Wahr. verw. Gebiete 29 39–44 Occurrence Handle0268.60004 Occurrence Handle362431 Occurrence Handle10.1007/BF00533185

    Article  MATH  MathSciNet  Google Scholar 

  • V.V. Yurinskii (1977) ArticleTitleOn the error of the Gaussian approximation for convolutions Theory Prob. Applications 2 236–247

    Google Scholar 

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Authors and Affiliations

  1. Depatament de Mathematique, Université Montpellier 2, 34095, Montpellier Cedex 5, France

    André Mas

  2. Laboratoire de Probabilités-Statistiques, Université Montpellier 2, CC051, Place Eugène Bataillon, 34095, Montpellier Cedex 5

    André Mas

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In final form November 2004

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Mas, A. Testing for the Mean of Random Curves: A Penalization Approach. Stat Infer Stoch Process 10, 147–163 (2007). https://doi.org/10.1007/s11203-005-0754-3

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