Random designs for robustness to functional model misspecification

Tim Waite, School of Mathematics, University of Manchester

Joint work with Dave Woods, Southampton Statistical Sciences Research Institute, University of Southampton

Statistical design of experiments allows empirical studies in science and engineering to be conducted more efficiently through careful choice of the settings of the controllable variables under investigation. Much conventional work in optimal design of experiments begins by assuming a particular structural form for the model generating the data, or perhaps a small set of possible parametric models. However, these parametric models will only ever be an approximation to the true relationship between the response and controllable variables, and the impact of this approximation step on the performance of the design is rarely quantified.

We consider response surface problems where it is explicitly acknowledged that a linear model approximation differs from the true mean response by the addition of a discrepancy function. The most realistic approaches to this problem develop optimal designs that are robust to discrepancy functions from an infinite-dimensional class of possible functions. Typically it is assumed that the class of possible discrepancies is defined by a bound on either (i) the maximum absolute value, or (ii) the squared integral, of all possible discrepancy functions.

Under assumption (ii), minimax prediction error criteria fail to select a finite design. This occurs because all finitely supported deterministic designs have the problem that the maximum, over all possible discrepancy functions, of the integrated mean squared error of prediction (IMSEP) is infinite.

We demonstrate a new approach in which finite designs are drawn at random from a highly structured distribution, called a designer, of possible designs. If we also average over the random choice of design, then the maximum IMSEP is finite. We develop a class of designers for which the maximum IMSEP is analytically and computationally tractable. Algorithms for the selection of minimax efficient designers are considered, and the inherent bias-variance trade-off is illustrated.