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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.

Methods for the construction of dose–response designs are presented that are robust against possible model misspecifications and mislabelled responses. The asymptotic properties are studied, leading to asymptotically minimax designs that minimize the maximum – over neighbourhoods of both types of model inadequacies – value of the mean squared error of the predictions. Both sequential and adaptive approaches are studied. Finite sample simulations and examples illustrate the gains to be made by adaptivity.

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 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.

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

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.

Experimentation in scientific or medical studies is often carried out in order to model the 'success' probability of a binary random variable. Experimental designs for the testing of lack of fit and for estimation, for data with binary responses depending upon covariates which can be controlled by the experimenter, are constructed. It is supposed that the preferred model is one in which the probability of the occurrence of the target outcome depends on the covariates through a link function (logistic, probit, etc.) evaluated at a regression response - a function of the covariates and of parameters to be estimated from the data, once gathered. The fit of this model is to be tested within a broad class of alternatives over which the regression response varies. To this end, the problem is phrased as one of discriminating between the preferred model and the class of alternatives. This, in turn, is a hypothesis testing problem, for which the asymptotic power of the test statistic is directly related to the Kullback-Leibler divergence between the models, averaged over the design. 'Maximin' designs, which maximize (through the design) the minimum (among the class of alternative models) value of this power together with a measure of the efficiency of the parameter estimates are also constructed. Several examples are presented in detail; two of these relate to a medical study of fluoxetine versus a placebo in depression patients. The method of design construction is computationally intensive, and involves a steepest descent minimization routine coupled with simulated annealing.

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 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.

Equality and proportionality of the ordinary least-squares estimator (OLSE), the weighted least-squares estimator (WLSE), and the best linear unbiased estimator (BLUE) for Xb in the general linear (Gauss–Markov) model M ¼ fy; Xb;s 2 Rg are investigated through the matrix rank method.

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.

In this article, we consider robust designs for approximate polynomial regression models, by applying the theory of canonical moments. The design criterion, first given in Liu and Wiens (J. Statist. Planning Inference 64 (1997) 369), is to maximize the determinant of the information matrix subject to a side condition of bounding the bias arising from model misspecification. We give a new proof of, and extend, the main theorem in Liu and Wiens (op. cit.); in so doing we shed new light on the structure of this problem. New designs, with the further property of minimizing the generalized variance of the additional regression coefficients when an enlarged model is fitted, are derived and assessed. These provide additional robustness against uncertainty regarding the proper degree of the fitted polynomial response.

We construct minimax robust designs for estimating wavelet regression models. Such models arise from approximating an unknown nonparametric response by a wavelet expansion. The designs are robust against errors in such an approximation, and against heteroscedasticity. We aim for exact, rather than approximate, designs; this is facilitated by our use of simulated annealing. The relative simplicity of annealing allows for a much more complete treatment of some hitherto intractable problems initially addressed in Oyet and Wiens (J. Nonparametric Stat. 12 (2000) 837). Thus, we are able to exhibit integer-valued designs for estimating higher order wavelet approximations of nonparametric curves. The exact designs constructed for multiwavelet approximations of various orders are found to be symmetric and periodic, as anticipated in Oyet and Wiens (J. Nonparametric Stat. 12 (2000) 837). We also construct integer-valued designs based on the Daubechies wavelet system with a wavelet number of 5.

The authors introduce the formal notion of an approximately specified nonlinear regression model and investigate sequential design methodologies when the fitted model is possibly of an incorrect parametric form. They present small-sample simulation studies which indicate that their new designs can be very successful, relative to some common competitors, in reducing mean squared error due to model misspecifi-cation and to heteroscedastic variation. Their simulations also suggest that standard normal-theory inference procedures remain approximately valid under the sequential sampling schemes. The methods are illustrated both by simulation and in an example using data from an experiment described in the chemical engineering literature. Les auteurs definissent formellement le concept de modele de regression non lineaire approxima-tif et proposentdes plans d'experience sequentiels pour les situations o4uG la forme parametrique du modele ajuste est inexacte. Ils presentent une etude de simulation qui montre que, pour de petits echantillons, leurs nouveaux plans sont largement preferables aux plans usuels en terme de reduction de I'erreur quadratique moyenne associee a rinadequation du modele et a l'heteroscedasticite. Leurs simulations montrent aussi que les procedures d'inference classiques associees au paradigme normal restent valables, a peu de choses pres, pour ces plans expeimentaux se'quentiels. La methodologie proposde est illustree par voie de simulation et au moyen d'une application concrete tiree de la pratique du genie chimique.

We obtain designs for linear regression models under two main departures from the classical assumptions: (1) the response is taken to be only approximately linear, and (2) the errors are not assumed to be independent, but to instead follow a first-order autoregressive process. These designs have the property that they minimize (a modification of) the maximum integrated mean squared error of the estimated response, with the maximum taken over a class of departures from strict linearity and over all autoregression parameters p, Ipl < 1, of fixed sign. Specific methods of implementation are discussed. We find that an asymptotically optimal procedure for AR(1) models consists of choosing points from that design measure which is optimal for uncorrelated errors, and then implementing them in an appropriate order.

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 study designs for the regression model E[Y|x] = Σp−1j = 0θjxj + xpΨ(x), where Ψ(x) is unknown but bounded in absolute value by a given function ϱ(x). This class of response functions models departures from an exact polynomial response. We consider the construction of designs which are robust, with respect to various criteria, as the true response varies over this class. The resulting designs are shown to compare favourably with others in the literature.

For the approximately linear model Yi,~ = /~z(xi) + n-1/2fn(xi) + el, with i.i.d, errors ei and fixed carriers z(xi), we establish the asymptotic normality of a generalized M-estimator of regression/scale. The estimator minimizes a weighted Huber-Dutter loss function. The function f,(x) contributes a bias term to the asymptotic normal distribution; apart from this term the estimator is C~-equivalent to the estimator obtained assuming the response to be exactly linear. Several estimate/design combinations are compared, in a simulation study. AMS classifications: Primary 62J05, 62F35; Secondary 62F1 l, 62F12

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 problem of robust M-estimation of a vector of regression parameters, when the errors are dependent. We assume a weakly stationary, but otherwise quite general dependence structure. Our model allows for the representation of the correlations of any time series of finite length. We first construct initial estimates of the regression, scale, and autocorrelation parameters. The initial autocorrelation estimates are used to transform the model to one of approximate independence. In this transformed model, final one-step M-estimates are calculated. Under appropriate assumptions, the regression estimates so obtained are asymptotically normal, with a variance-covariance structure identical to that in the case in which the autocorrelations are known a priori. The results of a simulation study are given. Two versions of our estimator are compared with the L1 -estimator and several Huber-type M-estimators. In terms of bias and mean squared error, the estimators are generally very close. In terms of the coverage probabilities of confidence intervals, our estimators appear to be quite superior to both the L1-estimator and the other estimators. The simulations also indicate that the approach to normality is quite fast. RESUME Nous considerons le probleme de M-estimation robuste pour un vecteur de parametres de regression, lorsque les erreurs sont dependantes. Nous supposons une stationnarite faible, mais autrement une structure de dependance plutǒt generate. Notre modele permet la representation des correlations de n'importe quelle serie chronologique de longueur finie. Tout d'abord, nous construisons des estimateurs initiaux des parametres de regression, d'echelle et d'autocorrelation. Les estimateurs initiaux d'autocorrelation sont utilises afin de transformer le modele en un modele avec independence approximative. Les M-estimateurs finaux sont calcules avec ce modele transforme. Sous des hypotheses appropriees, les estimateurs de regression ainsi obtenus sont asymptotiquement normaux, avec une structure de variance/covariance identique a celle lorsque les autocorrelations sont connues a priori. Nous donnons les resultats d'une etude de simulation. Nous comparons deux versions de notre estimateur avec l'estimateur L1 et plusieurs M-estimateurs de type Huber. Les estimateurs sont generalement tres proches sur le plan du biais et de L'erreur quadratique moyenne. Nos estimateurs sont superieurs a l'estimateur L1 et aux autres estimateurs sur le plan des probabilites de couverture des interval les de confiance. Nos simulations indiquent aussi que la convergence vers la normale est tres rapide.

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 ask whether or not the saddlepoint property holds, for robust M-estimation of scale, in gross-errors and Kolmogorov neighbourhoods of certain distributions. This is of interest since the saddlepoint property implies the minimax property — that the supremum of the asymptotic variance of an M-estimator is minimized by the maximum likelihood estimator for that member of the distributional class with minimum Fisher information. Our findings are exclusively negative — the saddlepoint property fails in all cases investigated.

Badly weighted least squares—solution. An excellent solution was independently proposed by Geert Dhaene. A proof for the case p = 1 based on the convexity of Laplace transforms was also provided by Koenker and Portnoy, the posers of the problem.

We revisit a proposal, for robust sequential design in the presence of uncertainty about the regression response, previously made by these authors. We obtain conditions under which a sequence of designs for nonlinear regression models leads to asymptotically normally distributed estimates. The results are illustrated in a simulation study. We conclude that estimates computed after the experiment has been carried out sequentially may, in only moderately sized samples, be safely used to make standard normal-theory inferences, ignoring the dependencies arising from the sequential nature of the sampling. The quality of the normal approximation deteriorates somewhat when the random errors are heteroscedastic.

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived. Cet article concerne l'estimation robuste de parametres de position et de matrices de dispersion dans les situations ou la loi des erreurs est totalement inconnue sauf pour la presence d'une symetrie spherique a l'interieur d'une region centrale. On formule le modele de faon precise et on en analyse les parametres identifiables pour lesquels on construit des estimateurs initiaux convergents. A partir de ces estimations initiales, un processus iteratif nous permet de deduire des M-estimateurs convergents et asymptotiquement normaux. Ceux-ci sont fondes sur des “fonctions d'influence” qui s'annulent en dehors de certains ensembles compacts. Enfin, on obtient des M-estimateurs qui sont asymptotiquement minimax au sens de Huber.

We obtain the Mellin transform of the Beta statistic used in the test of the univariate general linear hypothesis, assuming that the error distribution is spherically symmetric. From this, the non-null distribution of the statistic is obtained. The normal-errors representation of the Beta as a central Beta with random d.f. is shown to hold iff the error distribution is a normal scale mixture. Closed form expressions for the density are given, without employing this assumption.

New partial orderings of life distributions are given. The concepts of decreasing mean residual life, new better than used in expectation, harmonic new better than used in expectation, new better than used in failure rate, and new better than used in failure rate average are generalized, so as to compare the aging properties of two arbitrary life distributions.

SUMMARY Multiplicative correction factors are derived for the limiting F distributions of two test statistics for parameter subsets in nonlinear regression. The factors depend on the first and second derivatives of the model and are related to measures of intrinsic nonlinearity. An example is given for which the correction is substantial. A similar factor is obtained for the lack of fit test in nonlinear regression.