Comparing search and estimation performances of designs based on a compound criterion

Since the introduction of the search design by Srivastava [Designs for searching non-negligible effects. In: Srivastava, editor. A survey of statistical design and linear models. Amsterdam: North-Holland, Elsevier; 1975. p. 507–519], construction of such designs has been considered by many researchers. The efficient performances of constructed search designs in terms of parameter estimation and search ability of parameters have also been investigated by several authors. They have proposed suitable optimality measures such as DD- and AD-optimality for estimation in the early stage of search design construction. Moreover, since 1990s, some criteria have been developed to evaluate search performance of a design. Although these criteria are useful none of them is able to evaluate both estimation and search efficiency of a design simultaneously. In this paper, we propose dual-task criteria to deal with searching and estimating performances of search designs. These compound criteria are weighted multiplication of estimation and search suitable criteria. They will be used for design comparison and the results will be presented.     

Secondary design considerations for minimum bias estimation

Several authors have suggested the method of minimum bias estimation for estimating response surfaces. The minimum bias estimation procedure achieves minimum average squared bias of the fitted model without depending on the values of the unknown parameters of the true surface. The only requirement is that the design satisfies a simple estimability condition. Subject to providing minimum average squared bias, the minimum bias estimator also provides minimum average variance of ?(x) where ?(x) is the estimate of the response at the point x.

To support the estimation of the parameters in the fitted model, very little has been suggested in the way of experimental designs except to say that a full rank matrix X of independent variables should be used. This paper presents a closer look at the estimability conditions that are required for minimum bias estimation, and from the form of the matrix X, a formula is derived which measures the amount of design flexibility available. The design flexibility is termed “the degrees of freedom” of the X matrix and it is shown how the degrees of freedom can be used to decide if other design optimality criteria might be considered along with minimum bias estimation. Several examples are provided.     

Optimal estimating function for estimation and prediction in semi-parametric models

In a seminal paper, Godambe [1985. The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428.] introduced the ‘estimating function’ approach to estimation of parameters in semi-parametric models under a filtering associated with a martingale structure. Later, Godambe [1987. The foundations of finite sample estimation in stochastic processes II. Bernoulli, Vol. 2. V.N.V. Science Press, 49–54.] and Godambe and Thompson [1989. An extension of quasi-likelihood Estimation. J. Statist. Plann. Inference 22, 137–172.] replaced this filtering by a more flexible conditioning. Abraham et al. [1997. On the prediction for some nonlinear time-series models using estimating functions. In: Basawa, I.V., et al. (Eds.), IMS Selected Proceedings of the Symposium on Estimating Functions, Vol. 32. pp. 259–268.] and Thavaneswaran and Heyde [1999. Prediction via estimating functions. J. Statist. Plann. Inference 77, 89–101.] invoked the theory of estimating functions for one-step ahead prediction in time-series models. This paper addresses the problem of simultaneous estimation of parameters and multi-step ahead prediction of a vector of future random variables in semi-parametric models by extending the inimitable approach of 13 and 14. The proposed technique is in conformity with the paradigm of the modern theory of estimating functions leading to finite sample optimality within a chosen class of estimating functions, which in turn are used to get the predictors. Particular applications of the technique give predictors that enjoy optimality properties with respect to other well-known criteria.     

Optimal experimental design criterion for discriminating semiparametric models

This paper studies the optimal experimental design problem to discriminate two regression models. Recently, López-Fidalgo et al. [2007. An optimal experimental design criterion for discriminating between non-normal models. J. Roy. Statist. Soc. B 69, 231–242] extended the conventional T-optimality criterion by Atkinson and Fedorov [1975a. The designs of experiments for discriminating between two rival models. Biometrika 62, 57–70; 1975b. Optimal design: experiments for discriminating between several models. Biometrika 62, 289–303] to deal with non-normal parametric regression models, and proposed a new optimal experimental design criterion based on the Kullback–Leibler information divergence. In this paper, we extend their parametric optimality criterion to a semiparametric setup, where we only need to specify some moment conditions for the null or alternative regression model. Our criteria, called the semiparametric Kullback–Leibler optimality criteria, can be implemented by applying a convex duality result of partially finite convex programming. The proposed method is illustrated by a simple numerical example.     

Optimal design for quasi-likelihood estimation in Poisson regression with random coefficients

Count data may be described by a Poisson regression model. If random coefficients are involved, maximum likelihood is not feasible and alternative estimation methods have to be employed. For the approach based on quasi-likelihood estimation a characterization of design optimality is derived and optimal designs are determined numerically for an example with random slope parameters.     

An optimality criterion for vector unbiased statistical estimation functions

Chandrasekar and Kale (1984) considered the problem of estimating a vector interesting parameter in the presence of nuisance parameters through vector unbiased statistical estimation functions (USEFs) and obtained an extension of the Cramér-Rao inequality. Based on this result, three optimality criteria were proposed and their equivalence was established. In this paper, motivated by the uniformly minimum risk criterion (Zacks, 1971, p. 102) for estimators, we propose a new optimality criterion for vector USEFs in the nuisance parameter case and show that it is equivalent to the three existing criteria.     

General orthogonal designs for parameter estimation of ANOVA models under weighted sum-to-zero constraints

ABSTRACT

Traditional studies on optimal designs for ANOVA parameter estimation are based on the framework of equal probabilities of appearance for each factor's levels. However, this premise does not hold in a variety of experimental problems, and it is of theoretical and practical interest to investigate optimal designs for parameters with unequal appearing odds. In this paper, we propose a general orthogonal design via matrix image, in which all columns’ matrix images are orthogonal with each other. Our main results show that such designs have A- and E-optimalities on the estimation of ANOVA parameters which have unequal appearing odds. In addition, we develop two simple methods to construct the proposed designs. The optimality of the design is also validated by a simulation study.     

A new criterion of confidence set estimation: Improvement of the Neyman shortness

In past studies various criteria have been proposed for evaluating the performance of a confidence set. However, each of these criteria often causes some unsatisfactory results even for the standard models such as location model, scale model and multinormal model. In this article, we propose a new criterion so that the procedure of the confidence set estimation based on the criterion can lead to a desirable confidence set at least for the above models. The approach is on the basis of an improvement of the Neyman shortness according to two steps. The first step is some kind of theoretical improvement, referring to a proposal of Pratt. As a result, we get a solution to Pratt's paradox. In the second step, we adopt a kind of robust or minimax procedure without sticking to the uniform optimality. In conclusion, it is shown that the procedure based on our criterion produces a desirable and acceptable confidence set.     

Optimal design for the proportional odds model

The authors construct locally optimal designs for the proportional odds model for ordinal data. While they investigate the standard D‐optimal design, they also investigate optimality criteria for the simultaneous estimation of multiple quantiles, namely D_A ‐optimality and the omnibus criterion. The design of experiments for the simultaneous estimation of multiple quantiles is important in both toxic and effective dose studies in medicine. As with c‐optimality in the binary response problem, the authors find that there are distinct phase changes when exploring extreme quantiles that require additional design points. The authors also investigate relative efficiencies of the criteria.     

Unbiased statistical estimation functions for parameters in presence of nuisance parameters

We consider the problem of estimating a vector interesting parameter in the presence of nuisance parameters through vector unbiased statistical estimation functions (USEFs). An extension of the Cramer—Rao inequality relevant to the present problem is obtained. Three possible optimality criteria in the class of regular vector USEFs are those based on (i) the non-negative definiteness of the difference of dispersion matrices (ii) the trace of the dispersion matrix and (iii) the determinant of the dispersion matrix. We refer to these three criteria as M-optimality, T- optimality and D-optimality respectively. The equivalence of these three optimality criteria is established. By restricting the class of regular USEFs considered by Ferreira (1982), we study some interesting properties of the standardized USEFs and establish essential uniqueness of standardized M-optimal USEF in this restricted class. Finally some illustrative examples are included.     

Response surface design evaluation and comparison

Designing an experiment to fit a response surface model typically involves selecting among several candidate designs. There are often many competing criteria that could be considered in selecting the design, and practitioners are typically forced to make trade-offs between these objectives when choosing the final design. Traditional alphabetic optimality criteria are often used in evaluating and comparing competing designs. These optimality criteria are single-number summaries for quality properties of the design such as the precision with which the model parameters are estimated or the uncertainty associated with prediction. Other important considerations include the robustness of the design to model misspecification and potential problems arising from spurious or missing data. Several qualitative and quantitative properties of good response surface designs are discussed, and some of their important trade-offs are considered. Graphical methods for evaluating design performance for several important response surface problems are discussed and we show how these techniques can be used to compare competing designs. These graphical methods are generally superior to the simplistic summaries of alphabetic optimality criteria. Several special cases are considered, including robust parameter designs, split-plot designs, mixture experiment designs, and designs for generalized linear models.     

Classification of three-level strength-3 arrays

Generalized aberration (GA) is one of the most frequently used criteria to quantify the suitability of an orthogonal array (OA) to be used as an experimental design. The two main motivations for GA are that it quantifies bias in a main-effects only model and that it is a good surrogate for estimation efficiencies of models with all the main effects and some two-factor interaction components. We demonstrate that these motivations are not appropriate for three-level OAs of strength 3 and we propose a direct classification with other criteria instead. To illustrate, we classified complete series of three-level strength-3 OAs with 27, 54 and 81 runs using the GA criterion, the rank of the matrix with two-factor interaction contrasts, the estimation efficiency of two-factor interactions, the projection estimation capacity, and a new model robustness criterion. For all of the series, we provide a list of admissible designs according to these criteria.     

DT-optimum designs for model discrimination and parameter estimation

The paper introduces DT-optimum designs that provide a specified balance between model discrimination and parameter estimation. An equivalence theorem is presented for the case of two models and extended to an arbitrary number of models and of combinations of parameters. A numerical example shows the properties of the procedure. The relationship with other design procedures for parameter estimation and model discrimination is discussed.     

A note on all-bias designs with applications in spline regression models

If a model is fitted to empirical data, bias can arise from terms which are not incorporated in the model assumptions. As a consequence the commonly used optimality criteria based on the generalized variance of the estimator of the model parameters may not lead to efficient designs for the statistical analysis. In this note some general aspects of all-bias designs are presented, which were introduced in this context by Box and Draper (1959). Using an interesting correspondence between the points of all-bias designs and the knots of quadrature formulas we establish sufficient conditions such that a given design is an all-bias design. The results are illustrated in the special case of spline regression models. In particular our results generalize recent findings of Woods and Lewis (2006).     

Bayesian optimal sequential design for nonparametric regression via inhomogeneous evolutionary MCMC

We develop a novel computational methodology for Bayesian optimal sequential design for nonparametric regression. This computational methodology, that we call inhomogeneous evolutionary Markov chain Monte Carlo, combines ideas of simulated annealing, genetic or evolutionary algorithms, and Markov chain Monte Carlo. Our framework allows optimality criteria with general utility functions and general classes of priors for the underlying regression function. We illustrate the usefulness of our novel methodology with applications to experimental design for nonparametric function estimation using Gaussian process priors and free-knot cubic splines priors.     

A generalisation of T‐optimality for discriminating between competing models with an application to pharmacokinetic studies

The T‐optimality criterion is used in optimal design to derive designs for model selection. To set up the method, it is required that one of the models is considered to be true. We term this local T‐optimality. In this work, we propose a generalisation of T‐optimality (termed robust T‐optimality) that relaxes the requirement that one of the candidate models is set as true. We then show an application to a nonlinear mixed effects model with two candidate non‐nested models and combine robust T‐optimality with robust D‐optimality. Optimal design under local T‐optimality was found to provide adequate power when the a priori assumed true model was the true model but poor power if the a priori assumed true model was not the true model. The robust T‐optimality method provided adequate power irrespective of which model was true. The robust T‐optimality method appears to have useful properties for nonlinear models, where both the parameter values and model structure are required to be known a priori, and the most likely model that would be applied to any new experiment is not known with certainty. Copyright © 2012 John Wiley & Sons, Ltd.     

Concordance-based estimation approaches for the optimal sufficient dimension reduction score

To characterize the dependence of a response on covariates of interest, a monotonic structure is linked to a multivariate polynomial transformation of the central subspace (CS) directions with unknown structural degree and dimension. Under a very general semiparametric model formulation, such a sufficient dimension reduction (SDR) score is shown to enjoy the existence, optimality, and uniqueness up to scale and location in the defined concordance probability function. In light of these properties and its single-index representation, two types of concordance-based generalized Bayesian information criteria are constructed to estimate the optimal SDR score and the maximum concordance index. The estimation criteria are further carried out by effective computational procedures. Generally speaking, the outer product of gradients estimation in the first approach has an advantage in computational efficiency and the parameterization system in the second approach greatly reduces the number of parameters in estimation. Different from most existing SDR approaches, only one CS direction is required to be continuous in the proposals. Moreover, the consistency of structural degree and dimension estimators and the asymptotic normality of the optimal SDR score and maximum concordance index estimators are established under some suitable conditions. The performance and practicality of our methodology are also investigated through simulations and empirical illustrations.     

Max–min optimal discriminating designs for several statistical models

In the literature, different optimality criteria have been considered for model identification. Most of the proposals assume the normal distribution for the response variable and thus they provide optimality criteria for discriminating between regression models. In this paper, a max–min approach is followed to discriminate among competing statistical models (i.e., probability distribution families). More specifically, k different statistical models (plausible for the data) are embedded in a more general model, which includes them as particular cases. The proposed optimal design maximizes the minimum KL-efficiency to discriminate between each rival model and the extended one. An equivalence theorem is proved and an algorithm is derived from it, which is useful to compute max–min KL-efficiency designs. Finally, the algorithm is run on two illustrative examples.     

Optimal designs for the logit and probit models for binary data

We consider optimal designs for a class of symmetric models for binary data which includes the common probit and logit models. We show that for a large group of optimality criteria which includes the main ones in the literature (e.g. A-, D-, E-, F- and G-optimality) the optimal design for our class of models is a two-point design with support points symmetrically placed about the ED50 but with possibly unequal weighting. We demonstrate how one can further reduce the problem to a one-variable optimization by characterizing various of the common criteria. We also use the results to demonstrate major qualitative differences between the F - and c-optimal designs, two design criteria which have similar motivation.     

Synthetic and composite estimation under a superpopulation model

Under a simple superpopulation model for an arbitrary sampling design we derive optimal linear unbiased estimators/predictors of a mean in a domain. They can be viewed as synthetic and composite estimators of small area estimation theory when no auxiliary variable is available. Moreover, we show that the only requirement for optimality of a sampling strategy is to use any sampling plan of fixed sample size together with traditional estimators (as designed for simple random sampling without replacement). Finally, for symmetric sampling plans, simplified formulas (based on the first two moments of sample sizes) for optimal synthetic and composite estimators and their MSE’s are derived. Throughout the paper we consistently use the model-design setup.