Krystyna Katulska

D-Optimal Chemical Balance Weighing Designs with N ≡ 0 (mod 4) and 3 Objects

In this article, the problem of estimation of the individual weights of three objects using a chemical balance weighing design is considered. We use the criterion of D-optimality. We assume that the covariance matrix of errors is the matrix of first-order autoregressive process. Such problems were discussed in Li and Yang (2005) and also in Yeh and Lo Huang (2005). We present some results of D-optimal designs in certain class of designs with the design matrix X ∈ M n×3(±1) such that each column of matrix X has at least one 1 and one −1.

On certain A-optimal chemical balance weighing designs

The estimation problem of individual weights of objects in the chemical balance weighing design using the criterion of A-optimality is studied. It is assumed that errors are correlated and have the same variances. The lower bound of the trace of the variance matrix of estimators is obtained and the necessary and sufficient conditions for this lower bound to be attained are proved. Some construction methods are given.

Optimum biased spring balance weighing designs

The problem of the estimation of the individual weights of objects in a biased spring balance weighing design has been studied. A lower bound for the variance of each of the estimated weights for a biased spring balance weighing design is obtained and a necessary and sufficient condition for this lower bound to be attained is given. Also some optimum biased spring balance weighing designs are found.

On Certain D-Optimal Spring Balance Weighing Designs

The estimation problem of individual weights of objects in spring balance weighing design using the criterion of D-optimality is discussed. It is assumed that variances of errors are not equal and errors are not correlated. The upper bound of the determinant of the information matrix of estimators is obtained and the conditions for this upper bound to be attained are proved. Some methods of constructions regular D-optimal spring balance weighing designs are demonstrated.

Optimum biased spring balance weighing designs with non-homogeneity of the variances of errors

Abstract The paper deals with the problem of estimating the individual weights of objects by using a biased spring balance weighing design with non-homogeneity of the variances of errors in the model. The lower bound for the variance of each of the estimated weights from this biased spring balance weighing design is obtained and a necessary and sufficient condition for attaining this lower bound is given. Also some optimum biased spring balance weighing designs are found.

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

SummaryThis 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.

D-optimal and highly D-efficient designs with non-negatively correlated observations

In this paper we consider D-optimal and highly D-efficient chemical balance weighing designs. The errors are assumed to be equally non-negatively correlated and to have equal variances. Some necessary and sufficient conditions under which a design is D*-optimal design (regular D-optimal design) are proved. It is also shown that in many cases D*-optimal design does not exist. In many of those cases the designs constructed by Masaro and Wong (2008) and some new designs are shown to be highly D-efficient. Theoretical results are accompanied by numerical search, suggesting D-optimality of designs under consideration.

Relations between Spring and Chemical Balance Weighing Designs with the Diagonal Covariance Matrix of Errors

The paper deals with the problem of estimating the individual weights of objects with minimum variances by using a weighing design with the diagonal covariance matrix of errors in the model. The necessary and sufficient conditions for optimum biased spring balance weighing designs with the diagonal covariance matrix of errors and for optimum chemical balance weighing designs with the diagonal covariance matrix of errors are given and the relations between these designs are investigated. Also new optimum weighing designs are found.