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In a previous article (Wiens, 1991) we established a maximin property, with respect to the power of the test for Lack of Fit, of the absolutely continuous uniform ‘design’ on a design space which is a subset of R q with positive Lebesgue measure. Here we discuss some issues and controversies surrounding this result. We find designs which maximize the minimum power, over a broad class of alternatives, in discrete design spaces of cardinality N . We show that these designs are supported on the entire design space. They are in general not uniform for fixed N , but are asymptotically uniform as N → ∞ . Several examples with N fixed are discussed; in these we find that the approach to uniformity is very quick.

We construct robust designs for nonlinear quantile regression, in the presence of both a possibly misspecified nonlinear quantile function and heteroscedasticity of an unknown form. The asymptotic mean-squared error of the quantile estimate is evaluated and maximized over a neighbourhood of the fitted quantile regression model. This maximum depends on the scale function and on the design. We entertain two methods to find designs that minimize the maximum loss. The first is local – we minimize for given values of the parameters and the scale function, using a sequential approach, whereby each new design point minimizes the subsequent loss, given the current design. The second is adaptive – at each stage, the maximized loss is evaluated at quantile estimates of the parameters, and a kernel estimate of scale, and then the next design point is obtained as in the sequential method. In the context of a Michaelis–Menten response model for an estrogen/hormone study, and a variety of scale functions, we demonstrate that the adaptive approach performs as well, in large study sizes, as if the parameter values and scale function were known beforehand and the sequential method applied. When the sequential method uses an incorrectly specified scale function, the adaptive method yields an, often substantial, improvement. The performance of the adaptive designs for smaller study sizes is assessed and seen to still be very favourable, especially so since the prior information required to design sequentially is rarely available.

We address problems of model misspecification in active learning. We suppose that an investigator will sample training input points (predictors) from a subpopulation with a chosen distribution, possibly different from that generating the underlying whole population. This is in particular justified when full knowledge of the predictors is easily acquired, but determining the responses is expensive. Having sampled the responses the investigator will estimate a, possibly incorrectly specified, regression function and then predict the responses at all remaining values of the predictors. We derive functions r(x) of the predictors x, and carry out probability weighted sampling with weights proportional to r(x). The functions r(·) are asymptotically minimax robust against the losses incurred by random measurement error in the responses, sampling variation in the inputs, and biases resulting from the model misspecification. In our applications the values of r(·) are functions of the diagonal elements of the “hat” matrix which features in a regression on the entire population; this yields an interpretation of sampling the “most influential” part of the population. Applications on simulated and benchmark data sets demonstrate the strong gains to be achieved in this manner, relative to passive learning and to previously proposed methods of active learning. We go on to illustrate the methods in the context of a case study relating ice thickness and snow depth at various locations in Canada, using a “population” of about 50,000 observations made available by Statistics Canada. The Canadian Journal of Statistics xx: 1–19; 2017 © 2017 Statistical Society of Canada

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´

We give methods for the construction of designs for linear models, when the purpose of the investigation is the estimation of the conditional quantile function and the estimation method is quantile regression. The designs are robust against misspecified response functions, and against unanticipated heteroscedasticity. The methods are illustrated by example, and in a case study in which they are applied to growth charts.

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.

The authors review and extend the literature on robust regression designs. Even for straight line regression, there are cases in which the optimally robust designs—in a minimax mean squared error sense, with the maximum evaluated as the “true” model varies over a neighbourhood of that fitted by the experimenter—have not yet been constructed. They fill this gap in the literature, and in so doing introduce a method of construction that is conceptually and mathematically simpler than the sole competing method. The technique used injects additional insight into the structure of the solutions. In the cases that the optimality criteria employed result in designs that are not invariant under changes in the design space, their methods also allow for an investigation of the resulting changes in the designs. The Canadian Journal of Statistics 41: 679–695; 2013 © 2013 Statistical Society of Canada Resume Les auteurs examinent la litterature en matiere de plans experimentaux robustes pour la regression et y contribuent. Meme pour la regression lineaire, il existe des cas ou les plans robustes optimaux –dans le sens d'erreur quadratique moyenne minimax dont le maximum evalue au >varie dans un voisinage de celui ajuste a l'origine –n'ont pas encore ete elabores. Les auteurs comblent cette lacune en presentant une methode d’elaboration plus simple, sur les plans conceptuel et mathematique, que la seule autre methode existante. La technique utilisee donne une nouvelle perspective sur la structure des solutions. Lorsque les criteres d'optimalite choisis menent a des plans qui ne sont pas invariants en presence de changements dans l'espace d’echantillonnage, les methodes proposees permettent aussi un examen de ces changements dans les plans. La revue canadienne de statistique 41: 679–695; 2013 © 2013 Societe statistique du Canada

Two robust estimators of a matrix-valued location parameter are introduced and discussed. Each is the average of the members of a subsample–typically of covariance or cross-spectrum matrices–with the subsample chosen to minimize a function of its average. In one case this function is the Kullback–Leibler discrimination information loss incurred when the subsample is summarized by its average; in the other it is the determinant, subject to a certain side condition. For each, the authors give an efficient computing algorithm, and show that the estimator has, asymptotically, the maximum possible breakdown point. The main motivation is the need for efficient and robust estimation of cross-spectrum matrices, and they present a case study in which the data points originate as multichannel electroencephalogram recordings but are then summarized by the corresponding sample cross-spectrum matrices. Fil: Lind, John C.. Alberta Hospital Edmonton; Canadá Fil: Wiens, Douglas P.. University of Alberta; Canadá Fil: Yohai, Victor Jaime. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina

We investigate methods for the design of sample surveys, and address the traditional resistance of survey samplers to the use of model-based methods by incorporating model robustness at the design stage. The designs are intended to be sufficiently flexible and robust that resulting estimates, based on the designer's best guess at an appropriate model, remain reasonably accurate in a neighbourhood of this central model. Thus, consider a finite population of N units in which a survey variable Y is related to a q dimensional auxiliary variable x. We assume that the values of x are known for all N population units, and that we will select a sample of n≤N population units and then observe the n corresponding values of Y. The objective is to predict the population total $T=\sum_{i=1}^{N}Y_{i}$ . The design problem which we consider is to specify a selection rule, using only the values of the auxiliary variable, to select the n units for the sample so that the predictor has optimal robustness properties. We suppose that T will be predicted by methods based on a linear relationship between Y--possibly transformed--and given functions of x. We maximise the mean squared error of the prediction of T over realistic neighbourhoods of the fitted linear relationship, and of the assumed variance and correlation structures. This maximised mean squared error is then minimised over the class of possible samples, yielding an optimally robust (`minimax') design. To carry out the minimisation step we introduce a genetic algorithm and discuss its tuning for maximal efficiency.

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 give a description of the steps leading to a robustification of a dose–response study. The original experiment was designed and described by Rosenberger and Grill (1997) and has since been discussed by several other groups of researchers. Our robustification consists of redesigning the experiment so as to build in flexibility over a range of possible link functions, including the logistic link assumed by the original experimenters. We consider as well an asymptotic Neyman–Pearson test of the validity of the assumed link function.Copyright © 2011 John Wiley & Sons, Ltd.

Summary We construct experimental designs for dose–response studies. The designs are robust against possibly misspecified link functions; for this they minimize the maximum mean-squared error of the estimated dose required to attain a response in 100p% of the target population. Here p might be one particular value—p = 0.5 corresponds to ED50-estimation—or it might range over an interval of values of interest. The maximum of the mean-squared error is evaluated over a Kolmogorov neighbourhood of the fitted link. Both the maximum and the minimum must be evaluated numerically; the former is carried out by quadratic programming and the latter by simulated annealing.

In this study we developed an individual tree height prediction model for quaking aspen ( Populus tremuloides Michx.) grown in boreal mixedwood forests in Alberta using the nonlinear mixed model (NLMM) approach. We examined the impacts of density, species composition, and top height on aspen height predictions. Statistically significant stand level variables were incorporated into the base height–diameter model to increase the predictive ability and accuracy of the model at both the population and subject-specific levels. Our analyses showed that top height and density impacted height growth, but species composition did not. More importantly, we found that the inclusion of additional variables into the base model, despite improving model fitting statistics on the modelling data, did not improve the model's predictive ability and accuracy when cross-validated and when tested on an independent testing data set. Under the NLMM framework the base model performed as well as or better than the expanded models that contained other stand level variables. This has important theoretical and practical implications because, other than for biological reasons, more accurate local tree height predictions for aspen can be achieved simply by using the base height–diameter model fitted with the NLMM approach without the inclusion of other variables.

SummaryWe study the construction of experimental designs, the purpose of which is to aid in the discrimination between two possibly non-linear regression models, each of which might be only approximately specified. A rough description of our approach is that we impose neighbourhood structures on each regression response and determine the members of these neighbourhoods which are least favourable in the sense of minimizing the Kullback–Leibler divergence. Designs are obtained which maximize this minimum divergence. Both static and sequential approaches are studied. We then consider sequential designs whose purpose is initially to discriminate, but which move their emphasis towards efficient estimation or prediction as one model becomes favoured over the other.

We present and derive the asymptotic properties of a certain Neyman-Pearson test, relevant for problems concerning discrimination between two competing models. We then study an application in which this test is used to assess the efficacy of designs, the purpose of which is to aid in discriminating between two nonlinear regression models.

We develop criteria that generate robust designs and use such criteria for the construction of designs that insure against possible misspecifications in logistic regression models. The design criteria we propose are different from the classical in that we do not focus on sampling error alone. Instead we use design criteria that account as well for error due to bias engendered by the model misspecification. Our robust designs optimize the average of a function of the sampling error and bias error over a specified misspecification neighbourhood. Examples of robust designs for logistic models are presented, including a case study implementing the methodologies using beetle mortality data.

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 designs for the extrapolation of a regression response to one point outside of the design space. The response function is an only approximately known function of a specified linear function. As well, we allow for variance heterogeneity. We find minimax designs and corresponding optimal regression weights in the context of the following problems: (P1) for nonlinear least squares estimation with homoscedasticity, determine a design to minimize the maximum value of the mean squared extrapolation error (MSEE), with the maximum being evaluated over the possible departures from the response function; (P2) for nonlinear least squares estimation with heteroscedasticity, determine a design to minimize the maximum value of MSEE, with the maximum being evaluated over both types of departures; (P3) for nonlinear weighted least squares estimation, determine both weights and a design to minimize the maximum MSEE; (P4) choose weights and design points to minimize the maximum MSEE, subject to a side condition of unbiasedness. Solutions to (P1)–(P4) are given in complete generality. Numerical comparisons indicate that our designs and weights perform well in combining robustness and efficiency. Applications to accelerated life testing are highlighted.

We study minimax robust designs for response prediction and extrapolation in biased linear regression models. We extend previous work of others by considering a nonlinear fitted regression response, by taking a rather general extrapolation space and, most significantly, by dropping all restrictions on the structure of the regressors. Several examples are discussed.

We consider the construction of designs for test-control field experiments, with particular attention being paid to the effects of spatial correlation between adjoining plots. In contrast to previous approaches, in which very specific correlation structures were modelled, we explicitly allow a degree of uncertainty on the part of the experimenter. While fitting a particular correlation structure—and variance structure and regression response—the experimenter is thought to be seeking protection against other possible structures in full neighbourhoods of these particular choices. Robustness, in a minimax sense, is obtained through a modification of the kriging estimation procedure, and through the assignment of treatments to field plots.

We address the problem of finding robust sampling designs for the esti- mation of a discrete time second-order stationary process when its autocorrelation function is only approximately specified and has a spectral density belonging to a neighbourhood of a specified 'base' density. The value of the stochastic process is predicted by the best - for the assumed autocorrelation function - linear unbiased predictor on the basis of a finite sample of observations. Following the approach of minimax robustness, we find the least favourable - in the sense of maximizing the average mean squared error (amspe) of these predictions - spectral density. We then obtain, through a genetic algorithm, robust sampling designs which minimize this maximum amspe. Several examples are discussed and assessed, on the basis of which we conclude that the robust designs can offer substantial protection against model errors, at a minimal cost in efficiency at the base model. The techniques are illustrated in a case study, using a series of interest in statistical climatology.

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.