Minimax Estimates in a Semiparametric Model

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 that not only allows for adequate modeling of the response but also yields computable, reliable estimates of β. In this article two classes of g's are considered: In the first class, g is Lipschitz, whereas in the second, a modified Lipschitz condition holds on the first derivative of g. For the case that β is known to be 0 and g is to be estimated, Legostaeva and Shiryayev (1971) and Sacks and Ylvisaker (1978) studied similar classes of functions; these classes have already proven useful in applications (see Knafl, Sacks, and Ylvisaker 1985; Knafl, Speigelman, Sacks, and Ylvisaker 1984). Minimax (with respect to mean squared error) linear estimates of β are derived that are easy to compute and converge at rate 1/√n. Simulations indicate that the estimators perform well, even when the sample size is small.