Minimax Estimates in a Semiparametric Model
Abstract The statistical analysis of multidimensional data has been greatly influenced by the widespread use of computer-intensive smoothing techniques. These techniques allow one to look at the dependency of a response on several covariates without actually specifying the exact form of that dependency. Nevertheless, when the response's dependency on some subset of the covariates can be assumed to be, say, linear, common sense dictates incorporating this assumption into the estimation technique. In the semiparametric model with covariates X and t, the response, Y, is modeled as the sum of three components: g(t) with g (unknown) lying in an infinite-dimensional space, a linear component Xβ, and random error. If β were the parameter of interest, with the function g treated as a nuisance parameter, the conservative statistician would be led to a minimax approach, with the estimate of β minimizing the worst that could happen over a specified class of functions. Of course, the class of functions should be one ...