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In this article we investigate the optimal design problem for some wavelet regression models. Wavelets are very flexible in modeling complex relations, and optimal designs are appealing as a means of increasing the experimental precision. In contrast to the designs for the Haar wavelet regression model (Herzberg and Traves 1994; Oyet and Wiens 2000), the I-optimal designs we construct are different from the D-optimal designs. We also obtain c-optimal designs. Optimal (D- and I-) quadratic spline wavelet designs are constructed, both analytically and numerically. A case study shows that a significant saving of resources may be realized by employing an optimal design. We also construct model robust designs, to address response misspecification arising from fitting an incomplete set of wavelets.

We present and discuss the theory of minimax I- and D-robust designs on a finite design space, and detail three methods for their construction that are new in this context: (i) a numerical search for the optimal parameters in a provably minimax robust parametric class of designs, (ii) a first-order iterative algorithm similar to that of Wynn (Ann Math Stat 5:1655–1664, 1970), and (iii) response-adaptive designs. These designs minimize a loss function, based on the mean squared error of the predicted responses or the parameter estimates, when the regression response is possibly misspecified. The loss function being minimized has first been maximized over a neighbourhood of the approximate and possibly inadequate response being fitted by the experimenter. The methods presented are all vastly more economical, in terms of the computing time required, than previously available algorithms.

MSC 2010: Primary 62K05; 62G35; secondary 62D05 Abstract: We address the resistance, somewhat pervasive within the sampling community, to model-based methods. We do this by introducing notions of "approximate models" and then deriving sampling methods which are robust to model misspecification within neighbourhoods of the sampler's approximate, working model. Specifically we study robust sampling designs for model-based stratification, when the assumed distribution F0 (·) of an auxiliary variable x, and the mean function and the variance function g0 (·) in the associated regression model, are only approximately specified. We adopt an approach of "minimax robustness," to which end we introduce neighbourhoods of the "working" F0 (·), and working regression model, and maximize the prediction mean squared error (mse) for the empirical best predictor, of a population total, over these neighbourhoods. Then we obtain robust sampling designs, which minimize an upper bound of the maximum mse through a modified genetic algorithm with "artificial implantation." The techniques are illustrated in a case study of Australian sugar farms, where the goal is the prediction of total crop size, stratified by farm size. The Canadian Journal of Statistics 43: 554-577; 2015 © 2015 Statistical Society of Canada R´ esumLes auteurs tentent de diminuer la rg´

To aid in the discrimination between two, possibly nonlinear, regression models, we study the construction of experimental designs. Considering that each of these two models might be only approximately specified, robust “maximin” designs are proposed. The rough idea is as follows. We impose neighbourhood structures on each regression response, to describe the uncertainty in the specifications of the true underlying models. We determine the least favourable—in terms of Kullback–Leibler divergence—members of these neighbourhoods. Optimal designs are those maximizing this minimum divergence. Sequential, adaptive approaches to this maximization are studied. Asymptotic optimality is established.

We obtain V-optimal designs, which minimize the average variance of predicted regression responses, over a finite set of possible regressors. We assume a general and possibly heterogeneous variance structure depending on the design points. The variances are either known (or at least reliably estimated) or unknown. For the former case we exhibit optimal static designs; our methods are then modified to handle the latter case, for which we give a sequential estimation method which is fully adaptive, yielding both consistent variance estimates and an asymptotically V-optimal design.

We outline the construction of robust, static designs for nonlinear regression models. The designs are robust in that they afford protection from increases in the mean squared error resulting from misspecifications of the model fitted by the experimenter. This robustness is obtained through a combination of minimax and Bayesian procedures. We first maximize (over a neighborhood of the fitted response function) and then average (with respect to a prior on the parameters) the sum (over the design space) of the mean squared errors of the predictions. This average maximum loss is then minimized over the class of designs. Averaging with respect to a prior means that there is no remaining dependence on unknown parameters, thus allowing for static, rather than sequential, design construction. The minimization over the class of designs is carried out by implementing a genetic algorithm. Several examples are discussed.

We study designs, optimal up to and including terms that are O(n ?1), for weighted least squares regression, when the weights are intended to be inversely proportional to the variances but are estimated with random error. We take a finite, but arbitrarily large, design space from which the support points are to be chosen, and obtain the optimal proportions of observations to be assigned to each point. Specific examples of D- and I-optimal design for polynomial responses are studied. In some cases the same designs that are optimal under homoscedasticity remain so for a range of variance functions; in others there tend to be more support points than are required in the homoscedastic case. We also exhibit minimax designs, that minimize the maximum, over finite classes of variance functions, value of the loss. These also tend to have more support points, often resulting from the breaking down of replicates into clusters.

We obtain designs which are optimally robust against possibly misspecified regression models, assuming that the parameters are to be estimated by one of several types of M-estimation. Such designs minimize the maximum mean squared error of the predicted values, with the maximum taken over a class of departures from the fitted response function. One purpose of the study is to determine if, and how, the designs change in response to the robust methods of estimation as compared to classical least squares estimation. To this end, numerous examples are presented and discussed.

We discuss optimal design problems for a popular method of series estimation in regression problems. Commonly used design criteria are based on the generalized variance of the estimates of the coefficients in a truncated series expansion and do not take possible bias into account. We present a general perspective of constructing robust and efficient designs for series estimators which is based on the integrated mean squared error criterion. A minimax approach is used to derive designs which are robust with respect to deviations caused by the bias and the possibility of heteroscedasticity. A special case results from the imposition of an unbiasedness constraint; the resulting ''unbiased designs'' are particularly simple, and easily implemented. Our results are illustrated by constructing robust designs for series estimation with spherical harmonic descriptors, Zernike polynomials and Chebyshev polynomials.

We consider the construction of locally D-optimal designs for a nonlinear, multistage model in which one observes a binary response variable with expected value P ( x ; θ ) = H ( θ 0 + θ 1 x + ⋯ + θ k x k ) . Here H is any twice differentiable distribution function. Our results apply as well to heteroscedastic polynomial regression models, under mild conditions on the efficiency function.

We investigate the problem of designing for linear regression models, when the assumed model form is only an approximation to an unknown true model, using two novel approaches. The first is based on a notion of averaging of the mean-squared error of predictions over a neighbourhood of contaminating functions. The other is based on the usual D-optimal criterion but subject to bias-related constraints in order to ensure robustness to model misspecification. Both approaches are integer-valued constructions in the spirit of Fang and Wiens [2000. Integer-valued, minimax robust designs for estimation and extrapolation in heteroscedastic, approximately linear models. J. Amer. Statist. Assoc. 95(451), 807-818]. Our results are similar to those that have been reported using a minimax approach even though the rationale for the designs presented here are based on the notion of averaging, rather than maximizing, the loss over the contamination space. We also demonstrate the superiority of an integer-valued construction over the continuous designs using specific examples. The designs which protect against model misspecification are clusters of observations about the points that would have been the design points for classical variance-minimizing designs.

We consider robust methods for the construction of sampling designs in spatial studies. The designs are robust against misspecified regression responses, and are tailored for possible use with predictors which are minimax robust against misspecified variance/covariance structures. The loss function is based on the mean squared error of the predicted values. This is maximized, analytically, over a neighbourhood quantifying the departures from the fitted linear regression response. This maximum is then minimized numerically—by simulated annealing, or sequentially—in order to obtain the optimal designs. Copyright © 2004 John Wiley & Sons, Ltd.

We present schemes for the allocation of subjects to treatment groups, in the presence of prognostic factors. The allocations are robust against incorrectly specified regression responses, and against possible heteroscedasticity. Assignment probabilities which minimize the asymptotic variance are obtained. Under certain conditions these are shown to be minimax (with respect to asymptotic mean squared error) as well. We propose a method of sequentially modifying the associated assignment rule, so as to address both variance and bias in finite samples. The resulting scheme is assessed in a simulation study. We find that, relative to common competitors, the robust allocation schemes can result in significant decreases in the mean squared error when the fitted models are biased, at a minimal cost in efficiency when in fact the fitted models are correct.

We consider the problem of obtaining static (i.e., nonsequential), approximate optimal designs for a nonlinear regression model with response E[Y|x] = exp(θ0 + θ1 x + · + θ k x k ). The problem can be transformed to the design problem for a heteroscedastic polynomial regression model, where the variance function is of an exponential form with unknown parameters. Under the assumption that sufficient prior information about these parameters is available, minimally supported Bayesian D-optimal designs are obtained. A general procedure for constructing such designs is provided; as well the analytic forms of these designs are derived for some special priors. The theory of canonical moments and the theory of continued fractions are applied for these purposes.

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression. Les auteurs proposent et explorent de nouveaux plans experimentaux pour la regression. Ces plans sont minimax par rapport a une classe parametrique restreinte et s'averent a la fois robustes au biais dǔ a un mauvais choix de modele et raisonnablement efficaces. L'introduction de cette classe restreinte de plans est motivee par le desir d'eviter les problemes mathematiques et numeriques lies a la theorie minimax generale. Les plans sont robustes a des familles de modeles suffisamment larges pour que les mesures des plans minimax soient absolument continues. Les exemples d'implantation concernent l'approximation polynomiale et la regression multiple du second ordre.

We introduce an infinitesimal approach to the construction of robust designs for linear models. The resulting designs are robust against small departures from the assumed linear regression response and/or small departures from the assumption of uncorrelated errors. Subject to satisfying a robustness constraint, they minimize the determinant of the mean squared error matrix of the least squares estimator at the ideal model. The robustness constraint is quantified in terms of boundedness of the Gateaux derivative of this determinant, in the direction of a contaminating response function or autocorrelation structure. Specific examples are considered. If the aforementioned bounds are sufficiently large, then (permutations of) the classically optimal designs, which minimize variance alone at the ideal model, meet our robustness criteria. Otherwise, new designs are obtained.

We consider the problem of the sequential choice of design points in an approximately linear model. It is assumed that the fitted linear model is only approximately correct, in that the true response function contains a nonrandom, unknown term orthogonal to the fitted response. We also assume that the parameters are estimated by M-estimation. The goal is to choose the next design point in such a way as to minimize the resulting integrated squared bias of the estimated response, to order n-1. Explicit applications to analysis of variance and regression are given. In a simulation study the sequential designs compare favourably with some fixed-sample-size designs which are optimal for the true response to which the sequential designs must adapt.

We obtain designs, to be used for investigations of response surfaces by regression techniques, when (i)the fitted, linear (in the parameters) response is incorrect and (ii)the parameters are to be estimated robustly. Minimax designs are determined for ‘small’ departures from the fitted response. We specialize to the case in which the experimenter fits a plane, when in fact the true response contains quadratic and interaction terms. In this case, minimax rotatable designs are derived, subject to a lower bound on the power of a robust test of model adequacy. The optimal designs place their mass at the centre of the design space, and on a sphere interior to the design space.

We consider the approximately linear regression model E[y|x] = z T (x)θ + ƒ(x), x∈ S , where ƒ(x) is a non-linear disturbance restricted only by a bound on its L 2(S) norm, and where S is the design space. For loss functions which are monotonic functions of the mean squared error matrix, we derive a theory to guide in the construction of designs which minimize the maximum (over ƒ ;) loss. We then specialize to the case zT (x) = (1, xT), so that the fitted surface is a plane. In this case we give minimax designs for loss functions corresponding to the classical D-, A-, E-, Q- and G-optimality criteria.

We exhibit regression designs and weights which are robust against incorrectly specified regression responses and error heteroscedasticity. The approach is to minimize the maximum integrated mean squared error of the fitted values, subject to an unbiasedness constraint. The maxima are taken over broad classes of departures from the `‘ideal’ model. The methods yield particularly simple treatments of otherwise intractable design problems. This point is illustrated by applying these methods in a number of examples including polynomial and wavelet regression and extrapolation. The results apply to generalized M-estimation as well as to least squares estimation. Two open problems - one concerning designing for polynomial regression and the other concerning lack of fit testing - are given.

In many experiments, not all explanatory variables can be controlled. When the units arise sequentially, different approaches may be used. The authors study a natural sequential procedure for "mar- ginally restricted" D-optimal designs. They assume that one set of explanatory variables (xi) is observed sequentially, and that the experimenter responds by choosing an appropriate value of the explanatory vari- able X2. In order to solve the sequential problem a priori, the authors consider the problem of constructing optimal designs with a prior marginal distribution for xi. This eliminates the influence of units already observed on the next unit to be designed. They give explicit designs for various cases in which the mean re- sponse follows a linear regression model; they also consider a case study with a nonlinear logistic response. They find that the optimal strategy often consists of randomizing the assignment of the values of X2. Plans sequentiels D-optimaux sous contraintes marginales Resume : Dans bien des experiences, on ne peut pas controler toutes les variables explicatives. Lorsque les unites arrivent sequentiellement, diverses approches sont possibles. Les auteurs etudient une procedure se- quentielle naturelle pour les plans D-optimaux sous "contraintes marginales." Ils supposent qu'un ensemble de variables explicatives (xi) est observe sequentiellement et que l'experimentateur reagit en choisissant un niveau approprie pour la variable explicative xi. Pour resoudre ce probleme sequentiel a priori, les auteurs proposent la construction de plans optimaux etant donne une loi a priori marginale pour xi. Ceci a pour