**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.