Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 3, pp. 452--461

 ASYMPTOTICS FOR M-TYPE REGRESSION SPLINES WITH AUXILIARY SCALE ESTIMATION

By

  XUMING HE, University of Illinois

and

PEIDE SHI, Peking University

SUMMARY.   We consider M-estimation of nonparametric regression based on B-spline approximations. Scale invariance of the M-type regression splines can be achieved by using a preliminary estimate scale.It is expected that the optimal rates of convergence for the M-type regression splines are preserved when the preliminary scale estimate is consistent at the same time or better rate. If the scale estimate is root-n consistent, this result is proved for a large class of robust score functions including non-differentiable ones like Huber's. For more general scale estimates, the result remains valid at least for sufficiently smooth score functions and for error distributions with a finite second moment.

AMS (1980) subject classification.   Primary 62G07, secondary 62G20, 62G35.

Key words and phrases. $B$-spline, $M$-estimator, nonparametric regression, rate of convergence, scale.

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