Budget constrained run orders in optimum design

When experiments are to be performed in a time sequence, the observed responses may be affected by an unknown time trend. Run orders that are optimally balanced for time trends usually involve huge costs due to the large number of factor level changes. Therefore, trend-free run orders can be of low practical value in view of economical considerations. Tack and Vandebroek (J. Statist. Plann. Inference 98 (2001) 293) recently presented a design algorithm to construct trend-resistant run orders that give maximal information per unit cost. In this paper, the latter approach is extended to the construction of trendresistant run orders for which the total cost has to be lower than a specified budget. A new design algorithm is proposed that offers the experimenter a general method for solving a wide range of practical design problems.     

Selecting the optimum k in ridge regression

Two ridge rules are proposed for selecting the optimal k in ridge regression . Since the sampling distribution of the proposed rules are mathematically in tractable , a Monte Carlo study is conducted to examine their statisticl properties . Numerical results of the simulations in dicate that the performance of ridge rules depends upon the risk function used. Nevertheless, one of the ridge rules does produce a smaller mean squared error than the least squares estimator with the probability greater than 0.57 for all situations.     

Multiplicative algorithms for computing optimum designs

We study a new approach to determine optimal designs, exact or approximate, both for the uncorrelated case and when the responses may be correlated. A simple version of this method is based on transforming design points on a finite interval to proportions of the interval. Methods for determining optimal design weights can therefore be used to determine optimal values of these proportions. We explore the potential of this method in a range of examples encompassing linear and non-linear models, some assuming a correlation structure and some with more than one design variable.     

A-optimal designs for optimum mixture in an additive quadratic mixture model

In the analysis of experiments with mixtures, quadratic models have been widely used. The optimum designs for the estimation of optimum mixing proportions in a quadratic mixture model have been studied by Pal and Mandal [Optimum designs for optimum mixtures. Statist Probab Lett. 2006;76:1369–1379] and Mandal et al. [Optimum mixture designs: a pseudo-Bayesian approach. J Ind Soc Agric Stat. 2008;62(2):174–182; Optimum mixture designs under constraints on mixing components. Statist Appl. 2008;6(1&2) (New Series): 189–205], using a pseudo-Bayesian approach. In this paper, a similar approach has been employed to obtain the A-optimal designs for the estimation of optimum proportions in an additive quadratic mixture model, proposed by Darroch and Waller [Additivity and interaction in three-component experiments with mixture. Biometrika. 1985;72:153–163], when the number of components is 3, 4 and 5. It has been shown that the vertices of the simplex are necessarily the support points of the optimum design, and the other support points include barycentres of depth at most 2.     

Non-existence of an optimum estimator in a class of ratio estimators

A number of estimators formulated in the field of the ratio method of estimation has been presented. A class of estimators encompassing these estimators is constructed. It is noted that an optimum estimator does not exist uniformly in this class. The “Optimum” so obtained reduces to the usual regression estimator.     

Multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost

In stratified sampling, usually the cost function is taken as a linear function of sample sizes n _h . In many practical situations, the linear cost function does not approximate the actual cost incurred adequately. For example, when the cost of travelling between the units selected in the sample within a stratum is significant, instead of a linear cost function, a cost function that is quadratic in √n _h will be a more close approximation to the actual cost. In this paper, the problem is formulated as multi-objective nonlinear integer programming problem with quadratic cost under three different situations, i.e. complete, partial or null information about the population. A numerical example is also presented to illustrate the computational details.     

Formulating robust regression estimation as an optimum allocation problem

The Mallows-type estimator, one of the most reasonable bounded influence estimators, often downweights leverage points regardless of the magnitude of the corresponding residual, and this could imply a loss of efficiency. In this article, we consider whether the efficiency of this bounded influence estimator could be improved by regarding both the robust x -distance and the residual size. We develop a new robust procedure based on the ideas of the Mallows-type estimator and the general robust recipe, where data been cleaned by pulling outliers towards their fitted values. Our basic idea is to formulate the robust estimation as an allocation problem, where the objective function is a Huber-type "loss" function, but the pulling resource is restricted. Using a mathematical programming technique, the pulling resource is optimally allocated to influential points <$>({x}_i, y_i)<$> with respect to residual size and given weights, <$>w({x}_i)<$>. Three previously published approaches are compared to our proposal via simulated experiments. In the case of contaminated data by regression outliers and "good" leverage points, the proposed robust estimator is a reasonable bounded influence estimator concerning both efficiency and norm of bias. In addition, the proposed approach offers the potential to establish constraints for the regression parameters and also may potentially provide insight regarding outlier detection.     

A simplified approach to determination of optimum conditions experiments

Response surface experimentation is an integral part of the development of a new process or product, but the relatively efficient statistical methodologies for such experimentation are underutilized by research and development scientists and engineers because of a lack of knowledge and/or understanding of these methodologies. To help to increase its utilization, a simplified approach to one such statistical methodology, known as the determination of optimum conditions, has been developed which can be used by scientists and engineers with a minimum of statistical knowledge.     

Maximum likelihood and optimum location of observation sites in geodetic and spatial surveying

In geodetic and spatial surveying, accurate statistical es-timates of the Cartesian coordinates of a stationary target depend, in general, on such factors as target position, number and type of instruments, their location, their accuracy, and their mix. Direct measurements are assumed to be made in elevation, azimuth, and range with errors independently and normally distributed. For a mix of instruments, theoretical arguments are established using maximum likelihood estimation to determine the optimal siting of the instruments which minimizes (GDOP)² or the square of the geo-metric dilution of precision = sum of the variances of the estimates in the three orthogonal directions.     

Some results on optimality in models with heteroscedastic errors from partial optimum designs

This paper studies optimum designs for linear models when the errors are heteroscedastic. Sufficient conditions are given in order to obtainD-, A- andE-optimum designs for a complete regression model from partial optimum designs for some sub-parameters. A result about optimality for a complete model from the optimality for the submodels is included. Supported by Junta de Andalucía, research group FQM244.     

Optimum designs for estimation of optimum point under cost constraint

In this paper, we consider the estimation of the optimum factor combination in a response surface model. Assuming that the response function is quadratic concave and there is a linear cost constraint on the factor combination, we attempt to find the optimum design using the trace optimality criterion. As the criterion function involves the unknown parameters, we adopt a pseudo-Bayesian approach to resolve the problem.     

Point estimation of a global optimum for large combinatorial problems

The basis for this paper is in the following observation: for a given “ intractable” optimization problem for which no efficient solution technique exists, if we can devise a systematic procedure for generating independent, heuristic solutions, we should be able to apply statistical extreme-value theory in order to obtain point estimates for the globally optimal solution. This observation has been mechanized in order to evaluate heuristic solutions and assess deviations from optimality, the strategy developed is applicable to a host of combinatorial problems. The assumptions of our model, along with computational experience are discussed.     

Asymptotically optimum estimation of a probability in inverse binomial sampling under general loss functions

The optimum quality that can be asymptotically achieved in the estimation of a probability p using inverse binomial sampling is addressed. A general definition of quality is used in terms of the risk associated with a loss function that satisfies certain assumptions. It is shown that the limit superior of the risk for p asymptotically small has a minimum over all (possibly randomized) estimators. This minimum is achieved by certain non-randomized estimators. The model includes commonly used quality criteria as particular cases. Applications to the non-asymptotic regime are discussed considering specific loss functions, for which minimax estimators are derived.     

Biased discriminant analysis: Evaluation of the optimum probability of misclassification

This article extends the work of DiPillo (1976) on the Biased Minimum x² Rule. The optimum value of k (the biasing factor) Is determined and the true probability of misclassification is found. The proportion improvements reported in the 1976 paper are shown to be conservative. Some suggestions for algorithms to determine the optimal value of k are presented.     

Kullback-leibler information approach to the optimum measurement point for bayesian estimation

When an appropriate parametric model and a prior distribution of its parameters are given to describe clinical time courses of a dynamic biological process, Bayesian approaches allow us to estimate the entire profiles from a few or even a single observation per subject. The goodness of the estimation depends on the measurement points at which the observations were made. The number of measurement points per subject is generally limited to one or two. The limited measurement points have to be selected carefully. This paper proposes an approach to the selection of the optimum measurement point for Bayesian estimations of clinical time courses. The selection is made among given candidates, based on the goodness of estimation evaluated by the Kullback-Leibler information. This information measures the discrepancy of an estimated time course from the true one specified by a given appropriate model. The proposed approach is applied to a pharmacokinetic analysis, which is a typical clinical example where the selection is required. The results of the present study strongly suggest that the proposed approach is applicable to pharmacokinetic data and has a wide range of clinical applications.     

Analytical approach for determining the cost optimum raw material quality level

In this paper, the expected cost of a raw

material quality characteristic is determined and

the cost optimum quality level is found. A series

of piecewise linear functions is used torepresent a general cost function. Examples are given in

which the distributions of quality characteristics are treated as being either

uniform or normal. The relationship between a raw

material characteristic and manufacturing cost is

assumed to be known

Determining cost optimum quality level Manitoba     

Central composite designs for estimating the optimum conditions for a second-order model

Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.     

Fuzzy multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost and parabolic fuzzy numbers

This article deals with the uncertainties in a multivariate stratified sampling problem. The uncertain parameters of the problem, such as stratum standard deviations, measurement costs, travel costs and total budget of the survey, are considered as parabolic fuzzy numbers and the problem is formulated as a fuzzy multi-objective nonlinear programming problem with quadratic cost function. Using α-cut, parabolic fuzzy numbers are defuzzified and then the compromise allocations of the problem are obtained by fuzzy programming for a prescribed value of α. To demonstrate the utility of the proposed problem a numerical example is solved with the help of [LINGO User?s Guid. Lindo Systems Inc., 1415 North Dayton Street, Chicago,Illinois-60622, (USA), 2013] software and the derived compromise optimum allocation is compared with deterministic and proportional allocations.