Estimating the direction in which a data set is most interesting

Summary Estimation of orientation is a key operation at each step in projection pursuit. Since projection pursuit is a nonparametric algorithm, and since even low-dimensional approximations to the target function must converge to their limits at rates considerably slower than n_-1²(where n is sample size), then it might be thought that the same is true of orientation estimates. It is shown in the present paper that this is not the case, and that estimation of orientation is a parametric operation, in the sense that, under mild nonparametric assumptions, correctly-chosen kernel-type orientation estimates converge to their limits at rate n_-1². This property is not enjoyed by standard projection pursuit orientation estimates, which converge at a slower rate than n_-1². Most attention in the present paper is focussed on the case of projection pursuit density approximation, but it is pointed out that our arguments hold generally. An important practical conclusion is that data should be smoothed less when estimating orientation than when constructing the final projection pursuit approximation.