Robust Designs For Approximately Linear Regression Models With Correlated Errors

In optimum design theory, many results have been obtained under the assumption of exactlycorrect response and homoscedasticity. But this assumption is not always correct. In most situ-ations, an unknown bias or contamination may exist between the assumed response and the trueresponse , so it's more dangerous to put the design obtained from the ideal model into practice. Toreduce the risk, we study the model-robust design problem for general models with an unknownbias or contamination and the correlated errors. We derive the criteria to obtain the robust optimaldesigns under different assumptions to the bias functions and the error structures and investigatethe robustness of the designs gotten from the criteria through numerical examples.In Chapter 2, we assume that the obtained observations consist of two parts, one is the trueresponse, the sum of the linear part and the bias part, which is expressed by approximately linearfunctions, the other is the random errors caused by incorrect observation. We suppose the true re-sponse function comes from a reproducing kernel Hilbert space and the errors are ?tted by the qthorder moving average process, MA(q). We emphasis the cases with q = 1 and q = 2. Taking theaverage expected quadratic loss for the least squares estimation as the loss function, we develop adesign criterion for getting robust designs by using a minimax method. We also prove the fact thatwhen the true response is ?tted by the tensor products of Hermite polynomial, our criterion has theorthogonal property. We give the calculated designs in diagrammatic representation. Giving thetrue variance-covariance matrix, we compare our designs with the classical designs obtained underthe iid assumption, and ef?ciencies are given in tables. The results show that all-bias design has aperfect approximation to the compounded designs and can be used when the weighted parameteris uncertainty. Similar to the conclusions of Wiens (1999) and Zhou (2001c), numerical resultsshow that the performance of our criteria has some relation to the signs of the process parametersand when a prior to them is elicited correctly, the optimal designs obtained through our criteriacan be better than the classical optimal designs.In Chapter 3, we study the model-robust design problem by applying the reproducing kernelspace approach. We still assume the observations formed by the true response and the randomerrors. In our model, the approximately linear function can be any real-valued function satisfyingsome conditions, the bias is unknown and from some class H with a probability measure P, andthe correlated errors are considered. We derive a design criterion in terms of the average expectedquadratic loss for generalized least squares estimation. In the section of numerical examples, wemainly investigate the examples in Sobolev-Hilbert space. We investigate two different structures.From the numerical examples, we ?nd that when the covariance matrix is known, the average...