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Sharp Lower Bound for Regression with Measurement Errors and Its Implication for Ill-Posedness of Functional Regression

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Abstract

Nonparametric regression estimation with Gaussian measurement errors in predictors is a classical statistical problem. It is well known that the errors dramatically slow down the rate of regression estimation, and this paper complement that result by presenting a sharp constant. Then an interesting example of using this sharp constant to discover a new curse of dimensionality in functional nonparametric regression is presented, and analysis of real data complements the theory.

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REFERENCES

  1. G. Aneiros, R. Cao, R. Fraiman, C. Genest, and P. Vieu, ‘‘Recent advances in functional data analysis and high-dimensional statistics,’’ J. Multiv. Anal. 170, 3–9 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Aneiros, S. Novo, and P. Vieu, ‘‘Variable selection in functional regression: A review,’’ J. Multiv. Anal. 188, 1–13 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Carroll, D. Ruppert, L. Stefanski, and C. Crainiceanu, C. Measurement Error in Nonlinear Models: A Modern Prospective (Chapman Hall, New York, 2006).

    Book  MATH  Google Scholar 

  4. R. Carroll, A. Delaigle, and P. Hall, ‘‘Nonparametric prediction in measurement error models,’’ JASA 104, 993–1014 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Cuevas, ‘‘A partial overview of the theory of statistics with functional data,’’ J. Statist. Plann. Infer. 147, 1–23 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Delaigle and A. Meister, ‘‘Nonparametric regression with heteroscedastic measurement errors-in-variables problem,’’ JASA 102, 1416–1426 (2007).

    Article  MATH  Google Scholar 

  7. S. Efromovich, Nonparametric Curve Estimation: Methods, Theory, and Applications (Springer, New York, 1999).

    MATH  Google Scholar 

  8. S. Efromovich, Missing and Modified Data in Nonparametric Estimation (Chapman Hall, Boca Raton, 2018).

    Book  MATH  Google Scholar 

  9. S. Efromovich, S. ‘‘On sharp nonparametric estimation of differentiable functions,’’ Statist. Probab. Letters 152, 9–14 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Fan, and Y. Truong, ‘‘Nonparametric regression with errors in variables,’’ An. Statist. 21, 1900–1925 (1993).

    MathSciNet  MATH  Google Scholar 

  11. J. Fan, and E. Masry, ‘‘Multivariate regression estimatin with errors-in-variables: Asymptotic normality for mixing processes,’’ Journal of Multivariare Anal. 43, 237–271 (1992).

    Article  MATH  Google Scholar 

  12. Y. Fan, G. James, and P. Radchenko, ‘‘Functional additive regression,’’ Ann. Statist. 43, 2296–2325 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Febrero-Bande, W. Gonzalez-Manteiga, and M. de la Fuente, Variable selection in functional additive regression models. Comput. Statist. 34, 469–487 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Ferraty, and P. Vieu, Nonparametric Functional Data Analysis Theory and Practice (Springer, New York, 2006).

    MATH  Google Scholar 

  15. J. Friedman, and W. Stuetzle, ‘‘Projection pursuit regression,’’ JASA 76, 817–823 (1981).

    Article  MathSciNet  Google Scholar 

  16. G. Geenens, ‘‘Curse of dimensionality and related issues in nonparametric functional regression,’’ Statist. Surveys 5, 30–43 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Gregory, E. Mammen, and M. Wahl, ‘‘Statistical inference in sparse high-dimensional additive models,’’ Ann. Statist. 49, 1514–1536 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Hall and P. Qiu, P. ‘‘Discrete-transform approach to deconvolution problems,’’ Biometrika 93, 135–148 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Hoffmann and O. Lepski, ‘‘Random rates in anisotropic regression,’’ Ann. Statist. 30, 325–396 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Horowitz, J. Klemela, and E. Mammen, ‘‘Optimal estimation in additive regression models,’’ Bernoulli 12, 271–298 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  21. Y. Ingster, ‘‘ Minimax esting nonparametric statistical hypotheses about density distribution in \(L_{p}\) metrics,’’ Theory Probab. Applic. 31, 384–389 (1986).

    Google Scholar 

  22. I. Johnstone, Gaussian Estimation: Sequence and Wavelet Models (Stanford, 2019).

  23. E. Lehmann, Theory of Point Estimation (Wiley, New York, 1983)

    Book  MATH  Google Scholar 

  24. A. Meister, Deconvolution Problems in Nonparametric Statistics (Springer, New York, 2009)

    Book  MATH  Google Scholar 

  25. J. Morris, ‘‘Functional regression,’’ Annual Rev. Statist. Appl. 2, 321–359 (2015).

    Article  Google Scholar 

  26. H.-G. Müller and F. Yao, ‘‘Functional additive models,’’ JASA 103, 1534–1544 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  27. H.-G. Müller, ‘‘Peter Hall, functional data analysis and random objects,’’ Ann. Statist. 44, 1867–1887 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer, Berlin, 1975).

    Book  Google Scholar 

  29. S. Novo, G. Aneiros, and P. Vieu, ‘‘Sparse semiparametric regression when predictors are mixture of functional and high-dimensional variables,’’ Test 30, 481–504 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Sadhanala, and R. Tibshirani, ‘‘Additive models with trend filtering,’’ Ann. Statist. 47, 3032–2068 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Schennach, ‘‘Nonparametric regression in the presence of measurement error,’’ Econom. Theory 20, 1046–1093 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  32. C. Stone, ‘‘Additive regression and other nonparametric models,’’ Ann. Statist. 13, 689–705 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  33. Z. Tan and C.-H. Zhang, ‘‘Doubly penalized estimation in additive regression with high-dimensional data,’’ Ann. Statist. 47, 2567–2600 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Tsybakov, Introduction to Nonparametric Estimation (Springer, New York, 2009).

    Book  MATH  Google Scholar 

  35. G. Yi, A. Delaigle, and P. Gustafson, (2021). Handbook of Measurement Error Models (CRC, Boca Raton, 2021).

    Book  MATH  Google Scholar 

  36. Z. Yin, F. Liu, and Y. Xie, ‘‘Nonparametric regression estimation with mixed measurement errors,’’ Appl. Mathem. 7, 2269–2284 (2016).

    Article  Google Scholar 

  37. J. Wang, J. Chiou, and H. Müller, ‘‘Functional data analysis,’’ Annual Rev. Statist. Applic. 3, 257–295 (2016).

    Article  Google Scholar 

  38. L. Wasserman, All About Nonparametric Statistics (Springer, New York, 2006).

    MATH  Google Scholar 

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ACKNOWLEDGMENTS

Valuable comments of the Editor-in-Chief, Prof. Balakrishnan, Associate Editor and the reviewers are greatly appreciated.

Funding

The research was supported by NSF grant DMS-1915845.

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Correspondence to Sam Efromovich.

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Efromovich, S. Sharp Lower Bound for Regression with Measurement Errors and Its Implication for Ill-Posedness of Functional Regression. Math. Meth. Stat. 32, 209–221 (2023). https://doi.org/10.3103/S1066530723030031

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  • DOI: https://doi.org/10.3103/S1066530723030031

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