Variance estimation for high-dimensional regression models

The paper is concerned with the problem of variance estimation for a high-dimensional regression model. The results show that the accuracy n(-1/2) of variance estimation can be achieved only under some restrictions on smoothness properties of the regression function and on the dimensionality of the model. In particular, for a two times differentiable regression function, the rate n(-1/2) is achievable only for dimensionality smaller or equal to 8. For a higher dimensional model, the optimal accuracy is n(-4/d) which is worse than n(-1/2). The rate optimal estimating procedure is presented. (C) 2001 Elsevier Science (USA).