| The problem of sequential design for a nonparametric regression with binary data is considered. The aim of the statistical analysis is the estimation of a quantal response curve p and of the percentiles of p. An adaptive method is developed that proposes the location of the next best design point on the basis of past observations. For the response curve estimation, theoretical results concerning the closeness of the sequential design to the optimal design are obtained. These results are backed by a simulation study. For the ED{dollar}alpha{dollar} estimation, the consistency of our estimator is derived by computing the MSE. Our sequential design performs very well in a simulation study for sample sizes up to 35. We observed that our method is better than the Robbins-Monro and Adaptive Robbins-Monro procedures and that, if the true model is not probit and logit, our method is better than Wu's logit-MLE method. A sequential design for estimating a quantal response surface in two dimensions is also proposed. Bias, variance and IMSE of kernel estimates are derived and the optimal design density w.r.t. IMSE is constructed. Since there is no clear definition of the percentile in two dimensions, we convert our problem into an optimization problem using the Voronoi Tesselation. Simulation results show that the proposed method performs very well. |
