Active learning algorithm using the maximum weighted log-likelihood estimator

We study the problems of constructing designs for the regression problems. Our aim is to estimate the mean value of the response variable. The distribution of the independent variable is appropriately chosen from among the continuous designs so as to decrease the integrated mean square error (IMSE) of the fitted values. When we use the design, we face obstacles such that the true regression function may not belong to the statistical model, that is the model is misspecified. In the case of misspecification, the estimation of the mean value of response variable by using the design has bias. We suggest a new method to construct the design which does not have the bias even when the statistical model is misspecified. On the standard construction of the design, the maximum log-likelihood estimator (mle) is used. On the other hand, we use the maximum weighted log-likelihood estimator (mwle). The design with mle increase the bias in the case of misspecification. The mwle corrects the mle and decreases the bias term of IMSE. We give some numerical experiments and illustrate the efficiency of the proposed methods. (C) 2002 Elsevier B.V. All rights reserved.