Douglas P. Wiens

Minimax designs for approximately linear regression

Abstract 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.

Integer-Valued, Minimax Robust Designs for Estimation and Extrapolation in Heteroscedastic, Approximately Linear Models

Abstract We present our findings on a new approach to robust regression design. This approach differs from previous investigations into this area in three respects: The use of a finite design space, the use of simulated annealing to carry out the numerical minimization problems, and in our search for integer-valued, rather than continuous, designs. We present designs for the situation in which the response is thought to be approximately polynomial. We also discuss the cases of approximate first- and second-order multiple regression. In each case we allow for possible heteroscedasticity and also obtain minimax regression weights. The results are extended to cover extrapolation of the regression response to regions outside of the design space. A case study involving dose-response experimentation is undertaken. The optimal robust designs, which protect against bias as well as variance, can be roughly described as being obtained from the classical variance-minimizing designs by replacing replicates with clusters of observations at nearby but distinct sites.

Partial orderings of life distributions with respect to their aging properties

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.

DIOPHANTINE REPRESENTATION OF THE SET OF PRIME NUMBERS

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. [4] [81 have proven that every recursively enumerable set is Diophantine, and hence that the set of prime numbers is Diophantine. From this, and work of Putnam [12], it follows that the set of prime numbers is representable by a polynomial formula. In this article such a prime representing polynomial will be exhibited in explicit form. We prove (in Section 2)

Designs for approximately linear regression: two optimality properties of uniform designs

We study regression designs, with an eye to maximizing the minimum power of the standard test for Lack of Fit. The minimum is taken over a broad class of departures from the assumed multiple linear regression model. We show that the uniform design is maximin. This design attains its optimality by maximizing the minimum bias in the regression estimate of [sigma]2. It is thus surprising that this same design has an optimality property relative to the estimation of [sigma]2 -- it minimizes the maximum bias, in a closely related class of departures from linearity.

Robust weights and designs for biased regression models : Least squares and generalized M-estimation

Abstract We consider an ‘approximately linear’ regression model, in which the mean response consists of a linear combination of fixed regressors and an unknown additive contaminant. Only the linear component can be modelled by the experimenter. We assume that the experimenter chooses design points and then estimates the regression parameters by weighted least squares or by generalized M-estimation. For this situation we exhibit designs and weights which minimize scalar-valued functions of the covariance matrix of the regression estimates, subject to a requirement of unbiasedness. We report the results of a simulation study in which these designs/weights result in significant improvements, with respect to both bias and mean squared error, when compared to some common competitors.

Minimax Robust Designs and Weights for Approximately Specified Regression Models With Heteroscedastic Errors

Abstract This article addresses the problem of constructing designs for regression models in the presence of both possible heteroscedasticity and an approximately and possibly incorrectly specified response function. Working with very general models for both types of departure from the classical assumptions, I exhibit minimax designs and correspondingly optimal weights. Simulation studies and a case study accompanying the theoretical results lead to the conclusions that the robust designs yield substantial gains over some common competitors, in the presence of realistic departures that are sufficiently mild so as to be generally undetectable by common test procedures. Specifically, I exhibit solutions to the following problems: P1, for ordinary least squares, determine a design to minimize the maximum value of the integrated mean squared error (IMSE) of the fitted values, with the maximum being evaluated over both types of departure; P2, for weighted least squares, determine both weights and a design to m...

Robust minimax designs for multiple linear regression

Abstract We establish an extension, to the case of multiple regression, of a result on minimax simple regression designs due to P. Huber. Designs are found which are minimax with respect to integrated mean squared error as the true response function varies over an L 2-neighbourhood of (1) a p-dimensional plane or (2) a bivariate surface with possible interactions between the regressors.

Minimax regression designs for approximately linear models with autocorrelated errors

We study the construction of regression designs, when the random errors are autocorrelated. Our model of dependence assumes that the spectral density g(~o) of the error process is of the form g(o)= (1 -a)go(~O ) + ~gl(o), where go(CO) is uniform (corresponding to uncorrelated errors), ct ~ [0, 1) is fixed, and gx(to) is arbitrary. We consider regression responses which are exactly, or only approximately, linear in the parameters. Our main results are that a design which is asymptotically (minimax) optimal for uncorrelated errors retains its optimality under autocorrelation if the design points are a random sample, or a random permutation, of points from this distribution. Our results are then a partial extension of those of Wu (Ann. Statist. 9 (1981), 1168-1177), on the robustness of randomized experimental designs, to the field of regression design.

Robust designs for approximately polynomial regression

Abstract 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.