Rates of convergence of some estimators for a semiparametric model

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Abstract
In this paper, we analyze the rates of convergence of the mean square error of a partial spline estimator and a Denby/Speckman - type estimator for the parametric component of a semiparametric model under a wide range of conditions. It is found that the Denby/Speckman - type estimator has a faster rate of convergence than the partial spline estimator for some cases. It is shown that the optimal rate of decrease for the smoothing parameter X under the criterion of minimizing MSE for the parametric component is not necessarily the same as the optimal rate for minimization of the predictive mean square error in the function estimate. Thus data based estimates for X optimal for predictive mean square error in the function, such as GCV, may not be optimal for mean square error in the parametric component, leaving open the question of a data based estimate for X in this context.