After stating the model and the design problem, we briefly present the results for regression design prior to the work of Kiefer and Wolfowitz. We then review the major results of Kiefer and Wolfowitz, particularly those on the theory of design, as well as the way the criterion has been extended to non-linear models. Finally, we discuss algorithms for constructing D-optimum designs.

D-Optimality for Regression Designs: A Review

Max, Timothy A.; Schreuder, Hans T.; Hazard, John W.; Oswald, Daniel D.; Teply, John; Alegria, Jim. 1996. The Pacific Northwest Region vegetation and inventory monitoring system. Res. Pap. PNW-RP-493. Portland, OR: U.S. Department of Agriculture, Forest Service, Pacific Northwest Research Station. 22 p. A grid sampling strategy was adopted for broad-scale inventory and monitoring of forest and range vegetation on National Forest System lands in the Pacific Northwest Region, USDA Forest Service. This paper documents the technical details of the adopted design and discusses alternative sampling designs that were considered. A less technical description of the selected design will be given elsewhere. The grid consists of a regular, square spacing with 5.47 kilometers (3.4 mi) between grid points. The primary sampling unit (PSU), established at each grid sampling point, consists of a circular, 1-hectare (2.47-acre) plot. The PSU is subsampled with a set of different-sized fixed-area subplots, as well as line transects, to assess all components of vegetation. The design is flexible and can be used with many types of maps. The theory of point and change estimation is described, as well as estimates of variation that assess the statistical precision of estimates.

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The Pacific Northwest Region vegetation and inventory monitoring system

Since the mid 1950s, there has been a well-developed theory of sample survey design inference embracing complex designs with stratification and unequal probabilities (Smith, 2001). Unequal probability sampling was first suggested by Hansen and Hurwitz (1943) in the context of sampling with replacement. Narain (1951), Horvitz and Thompson (1952) developed the corresponding theory for sampling without replacement. A large part of survey sampling literature is devoted to unequal probabilities sampling, and more than 50 sampling algorithms have been proposed.

Sampling with unequal probabilities - K. R. W. Brewer and M. Hanif.

We point out the relation between the theory of balanced arrays and various combinatorial areas of design of experiment. Recalling some combinatorial theorems from Srivastava [1971], we apply these to prove a class of new and simple but rather stringent results on the existence of balanced arrays. Applications to the special case of orthogonal arrays are also considered.

Balanced Arrays and Orthogonal Arrays

In this paper, we obtain balanced resolution V plans for 2m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal b...

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we present a class of fractional factorial designs of the 27 series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 27 factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).

Optimal balanced 2 7 fractional factorial designs of resolution V , with N ≦42

On a decision rule using dichotomies for identifying the nonnegligible parameter in certain linear models

A general ratio estimator of a population total is proposed as an approximation to the estimator introduced by Srivastava (1985,Bull. Internat. Statist. Inst.,51(10.3), 1–16). This estimator incorporates additional information gathered during the survey in a new way. Statistical properties of the general ratio estimator are given and its relationship to the estimator proposed by Srivastava is explored. A special kind of general ratio estimator is suggested and it turns out to be very efficient in a simulation study when compared to several other commonly used estimators.

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A general ratio estimator and its application in model based inference

On the costwise optimality of certain hierarchical and standard multiresponse models under the determinant criterion

: Consider the class of general incomplete multiresponse (GIM) designs in which the set of units is divided into blocks of equal size, such that in any block the same subset of responses is measured on each unit. It is shown that with respect to the trace criterion and a reasonable cost restriction, the subclass of hierarchical multiresponse (HM) designs is complete in the sense that given any GIM design, there exists a HM design such that the cost involved under the two designs is the same, but the trace of the covariance matrix of the estimates of the parameters under the HM design is less than or equal to the similar quantity under the GIM design. The results also establish the important fact that there is a large class of situations where the standard multiresponse model (under which all responses are measured on each unit) should not be used. The nonlinear programming problem associated with obtaining the optimum HM design is stated and solved. (Author)

J. N. Srivastava

Papers

Optimal balanced 2 7 fractional factorial designs of resolution V , with N ≦42

On a decision rule using dichotomies for identifying the nonnegligible parameter in certain linear models

A general ratio estimator and its application in model based inference

On the costwise optimality of certain hierarchical and standard multiresponse models under the determinant criterion

On the costwise optimality of hierarchical multiresponse randomized block designs under the trace criterion