R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.     

A note on component-wise kronecker designs

Summary The paper investigates the particular case of symmetrical block designs with block size two. It is shown that the component-wise Kronecker product of two such designs (c.f. Mukerjee 1981) is connected if and only if the numbers of treatments are relatively prime.     

A note on connectedness of block designs

Summary We give a simple proof for the well known result that a block design (for an additive, fixed effects block model withv treatments inb blocks) is connected iff its ℰ-matrix has rankv-1.     

A note on theE-optimality of block designs

Summary In this paper we consider the problem of determiningE-optimal block designs for experimental situations in whichv treatments are to be tested onn experimental units arranged inb blocks and where the block sizes and number of replications assigned to the treatments are allowed to vary. Some sufficient conditions are obtained for designs to beE-optimal in these situations and methods for constructing designs which satisfy the sufficient conditions given are also derived.     

A note on generalized aberration in factorial designs

In this paper we extend the wordlength pattern and minimum aberration for non-regular factorials. The new concepts, the generalized wordlength pattern and minimum generalized aberration, are proposed. Some connections between the generalized wordlength pattern and uniformity are given. Some applications of the new concepts in the Blackett and Burman's designs are discussed. Received: September 2000     

A note on the determination of A-optimal type block designs under fixed and mixed effects models

In this paper, we consider the problem of estimating a selected set of contrasts between v treatments using a block design consisting of b blocks of size k. Traditionally, the construction of A-optimal block designs for such situations has been carried out assuming a fixed effects model. In this paper, we show that A-optimal designs constructed under a fixed effects model are robust in the sense that these designs have maximal minimal efficiency when considered among all available designs and under all possible mixed effects models. AMS 1991 subject classifications: Primary 62K05; Secondary 62K10     

A note on randomized response models for quantitative data

Standard randomized response (RR) models deal primarily with surveys which usually require a yes or a no response to a sensitive question, or a choice for responses from a set of nominal categories. As opposed to that, Eichhorn and Hayre (1983) have considered survey models involving a quantitative response variable and proposed an RR technique for it. Such models are very useful in studies involving a measured response variable which is highly sensitive in its nature. Eichhorn and Hayre obtained an unbiased estimate for the expectation of the quantitative response variable of interest. In this note we propose a procedure which uses a design parameter (controlled by the experimenter) that generalizes Eichhorn and Hayres results. Such a procedure yields an estimate for the desired expectation which has a uniformly smaller variance.Acknowledgements We are grateful to two referees for their valuable and constructive comments.     

A note on autoregressive error components models

The purpose of this paper is to shed some light on the asymptotic behavior of a wide class of estimators for a dynamic error components model when only the number of individuals tends to infinity, the number of time periods being kept fixed. In particular, it is shown that this asymptotic behavior is highly dependent on the assumption about the initial observations and that it offers very good approximations to the small sample behavior of the various estimators under consideration.     

Optimal designs for linear mixture models

Summary In a recent paper S nee and M arquardt [8] considered designs for linear mixture models, where the components are subject to individual lower and/or upper bounds. When the number of components is large their algorithm XVERT yields designs far too extensive for practical purposes.
The purpose of this paper is to describe a numerical procedure resulting in a design of fixed size N , which is approximately D -optimal, and where the components may be subject to linear constraints (f.e. upper or lower bounds). The proposed method is more generally applicable for models linear in the independent variables and the parameters and the convex hull of the experimental region is a polyhedron whose vertices are known.     

A note on identification of multivariate time-series models

We analyze some of the difficulties of identifying multivariate time-series models with feedback, when carried through the two-stage procedure as in Haugh-Box (1977). We find that if the procedure is simplified, as in Jenkins (1979), the identified model can be seriously misspecified. When the procedure is applied correctly, it becomes extremely complicated, as in Granger-Newbold (1977). In such cases, there is a serious risk of overparametrization. The complication is mostly caused by the underlying structure of the multivariate model for the univariate innovations which can be considerably more complicated than the multivariate time-series model of interest.     

A note on individual choice models of migration

In this paper individual probabilistic choice models are developed for the decision to migrate and the choice among alternative destinations. The models are developed to investigate how characteristics of decision makers and of alternative destinations affect choice. Several migration models in the literature (e.g., mover-stayer models) are shown to be special cases, their behavioral assumptions are made explicit, and a framework for testing these assumptions is provided.     

A note on the balancing factors of gravity models

Application of the production constrained gravity-type spatial interaction model generally incorporates a “balancing of attractions” procedure to yield the fully constrained condition. Two techniques for balancing a singly constrained gravity model are compared in this note. These are the Federal Highways Administration adjusted attraction factor method and the Furness column and row balancing procedure. This comparison shows firstly that the two techniques are identical, and secondly that the balancing procedure results in an arbitrary distortion of the calibrated distribution function. Furthermore, some empirical results show that the balancing procedure does not necessarily improve the model's predictions on a cell by cell basis. These results indicate that balancing may be an unnecessary as well as undesirable step in the application of the singly constrained gravity model.     

A note on the robustness of Box-Behnken designs to the unavailability of data

Box-Behnken designs and central composite designs are efficient designs for fitting second order polynomials to response surfaces, because they use relatively small numbers of observations to estimate the parameters. In this paper we investigate the robustness of Box-Behnken designs to the unavailability of observations, in the sense of finding t _max , the maximum number of arbitrary rows in the design matrix that can be removed and still leave all of the parameters of interest estimable. The results are compared to the known results for the central composite designs found in MacEachern, Notz, Whittinghill & Zhu (1995). The blocked Box-Behnken designs are equally as robust as those that are not blocked. Received December 1997     

A note on the estimation of total weight in chemical balance weighing designs

Summary This paper studies the problem of estimation of the total weight of objects using a chemical balance weighing design under the restriction |L−R| ≤a, whereL andR represent the number of objects placed on the left and right pans, respectively. A lower bound for the variance of the estimated total weight is given and a necessary and sufficient condition for this lower bound to be attained is obtained. Finally, weighing designs for which this lower bound is attainable are constructed.     

Estimation and optimal designs for linear Haar-wavelet models

This paper gives an analytical expression for the best linear unbiased estimator (BLUE) of the unknown parameters in the linear Haar-wavelet model. From the analytical expression, we solve for the eigenvalues of the covariance matrix of the BLUE in analytical form. Further, we use these eigenvalues to construct some conventional discrete optimal designs for the model. The equivalences among these optimal designs are demonstrated and some examples are also given.      

Optimal and robust designs for trigonometric regression models

This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.     

Optimal designs under two related models for interference

We consider two related models for interference: one having different directional neighbour effects and one with the same neighbour effects. We show that optimal designs for one model can be obtained from optimal designs for the other model. Acknowledgement. We are grateful to H. Kushner for advance notice of his work. SP was supported by the Isfahan University of Technology, Iran.     

A note on the assumed distributions in stochastic frontier models

Stochastic frontier models all need an assumption on the distributional form of the (in)efficiency component. Generally this efficiency component is assumed to be half normally, truncated normally, or exponentially distributed. This paper shows that the exponential distribution is, just like the half normal distribution, a special case of the truncated normal distribution. Moreover, this paper discusses the implications that this finding has on estimation.     

A note on multi-step forecasting with functional coefficient autoregressive models

This paper presents and evaluates alternative methods for multi-step forecasting using univariate and multivariate functional coefficient autoregressive (FCAR) models. The methods include a simple “plug-in” approach, a bootstrap-based approach, and a multi-stage smoothing approach, where the functional coefficients are updated at each step to incorporate information from the time series captured in the previous predictions. The three methods are applied to a series of U.S. GNP and unemployment data to compare performance in practice. We find that the bootstrap-based approach out-performs the other two methods for nonlinear prediction, and that little forecast accuracy is sacrificed using any of the methods if the underlying process is actually linear.