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Asymptotics for M-type smoothing splines with non-smooth objective functions

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  • Ioannis Kalogridis

    (University of Leuven)

Abstract

M-type smoothing splines are a broad class of spline estimators that include the popular least-squares smoothing spline but also spline estimators that are less susceptible to outlying observations and model misspecification. However, available asymptotic theory only covers smoothing spline estimators based on smooth objective functions and consequently leaves out frequently used resistant estimators such as quantile and Huber-type smoothing splines. We provide a general treatment in this paper and, assuming only the convexity of the objective function, show that the least-squares (super-)convergence rates can be extended to M-type estimators whose asymptotic properties have not been hitherto described. We further show that auxiliary scale estimates may be handled under significantly weaker assumptions than those found in the literature and we establish optimal rates of convergence for the derivatives, which have not been obtained outside the least-squares framework. A simulation study and a real-data example illustrate the competitive performance of non-smooth M-type splines in relation to the least-squares spline on regular data and their superior performance on data that contain anomalies.

Suggested Citation

  • Ioannis Kalogridis, 2022. "Asymptotics for M-type smoothing splines with non-smooth objective functions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 373-389, June.
    • Handle: RePEc:spr:testjl:v:31:y:2022:i:2:d:10.1007_s11749-021-00782-y
    DOI: 10.1007/s11749-021-00782-y
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    References listed on IDEAS

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    1. Cunningham, J. K. & Eubank, R. L. & Hsing, T., 1991. "M-type smoothing splines with auxiliary scale estimation," Computational Statistics & Data Analysis, Elsevier, vol. 11(1), pages 43-51, January.
    2. Bai, Z. D. & Wu, Y., 1994. "Limiting Behavior of M-Estimators of Regression-Coefficients in High Dimensional Linear Models II. Scale-Invariant Case," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 240-251, November.
    3. Pollard, David, 1991. "Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory, Cambridge University Press, vol. 7(2), pages 186-199, June.
    4. Bai, Z. D. & Wu, Y., 1994. "Limiting Behavior of M-Estimators of Regression Coefficients in High Dimensional Linear Models I. Scale Dependent Case," Journal of Multivariate Analysis, Elsevier, vol. 51(2), pages 211-239, November.
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