Minimax optimal designs for nonparametric regression - A further optimality property of the uniform distribution

In the common nonparametric regression model Y-i,Y-n = g(t(i,n)) + sigma (t(i,n)) sigma (t(i,n)) epsilon (i), i = 1 ,..., n, with id.d. noise and nonrepeatable design points ti, we consider the problem of choosing an optimal design for the estimation of the mean function g. A minimax approach is adopted which searches for designs minimizing the maximum of the asymptotic integrated mean squared error, where the maximum is taken over an appropriately bounded class of functions (g, sigma). The minimax designs are found explicitly, and for certain special cases the optimality of the uniform distribution can be established.