• asymptotic risk;
  • hard thresholding;
  • kernel smoothing;
  • regression model;
  • semiparametric least squares;
  • simulation;
  • smooth thresholding


We consider a partially linear model in which the vector of coefficients β in the linear part can be partitioned as (β1, β2), where β1 is the coefficient vector for main effects (e.g. treatment effect, genetic effects) and β2 is a vector for ‘nuisance’ effects (e.g. age, laboratory). In this situation, inference about β1 may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables (Steinian shrinkage), or from dropping the nuisance variables if there is evidence that they do not provide useful information (pretesting). We investigate the asymptotic properties of Stein-type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein-type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein-type estimator outperforms the full model estimator uniformly. By contrast, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty-type estimator for partially linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty-type estimators. The comparison shows that the shrinkage method performs better than the absolute penalty-type estimation method when the dimension of the β2 parameter space is large.