Minimax estimators of regression functions under normalized quadratic loss functions and inequality restrictions

This paper deals with the estimation of a regression function f of unknown form by shrinked least squares estimates calculated on the basis of possibly replicated observa-tions, The estimation loss is chosen in a somewhat more realistic manner then the usual quadratic losses and is given by an adequately weighted sum of squared errors in estimat-ing the values of f at the design points, normalized by the squared norm of the regression function, Shrinked least squares (as special ridge estimators) have been proved by the suthors in special cases to be minimax under all estimatiors.

We investigate the shrinking of least squares estimators with the objective of minimiz-ing the least favourable risk. Here we assume a known lower bound for the magnitude of f and a known upper bound for the difference between f and some simple function approxi-mating f, e.g. we know that f is the sum of a quadratic polynomial and of some