SHI Peide
Journal of Systems Science and Complexity. 1994, 7(3): 270-281.
This paper deals with M-estimators for a semiparametric regression model Y=X^tβ_0 + g_0(T) + e，where Y is real-valued, T ranges over a nondegenerate compact interval, X ∈R^d, e is a random error, β_0 is a d-vector of parameters to be estimated, and go is an unknown smooth function whose mth derivative function satisfies a Holder condition with exponent γ∈（0, 1]. A B-spline is taken to approximate g_0, the M - estimators of β and go are defined, and their convergence rates are investigated. A Monte Carlo study is carried out. It is shown that when the random errors are normally distributed the M-estimators are as good as least square esthoators; however, when the random errors are drawn from a symmetrically contaminated normal distribution the M-estimators are superior to least square estmators; and when the random errors are distributed as Cauchy distribution the M-edimators seem acceptable but the least Square estimtors behave poorly. It is proved that the B-spline M-estimators of g_0 attain the convergence rate as that of the optimal global rate of convergence for nonparametric regression, and the M-estimators of β_0 attain the convergence rate n-1/2 under some conditions.
