Approximate standard errors in semiparametric models

We consider semiparametric models with p regressor terms and q smooth terms. We obtain an explicit expression for the estimate of the regression coefficients given by the back-fitting algorithm. The calculation of the standard errors of these estimates based on this expression is a considerable computational exercise. We present an alternative, approximate method of calculation that is less demanding. With smoothing splines, the method is exact, while with loess, it gives good estimates of standard errors. We assess the adequacy of our approximation and of another approximation with the help of two examples.