Selected Remarks on Highly D-efficient Spring Balance Weighing Designs

Here, we consider a new construction method of determining highly D‐efficient spring balance weighing designs in classes in which a D‐optimal design does not exist. We give some condi‐ tions determining the relations between the parameters of such designs and construction examples.


Introduction
Let us consider is called a design matrix of the spring balance weighing design. Originally, the name spring balance weighing design pertained to experiments connected with determining unknown weights of objects by the use of balance with one pan which is called a spring balance. Nowadays, such designs are applied in many branches of knowledge including economic survey, see Banerjee (1975), Ceranka and Graczyk (2014). Some aspects of applications of spring balance weighing designs in agriculture are given by Ceranka and Katulska (1987a;1987b;, and Graczyk (2013). The example of application of such designs in bioengineering is presented in Gawande and Patkar (1999). Various problems related to spring balance weighing designs are presented in the literature. They are focused on the optimality criteria of such designs. The classic works here are Jacroux and Notz (1983), Koukouvinos (1996). Another group of issues is concerned with determining new methods of construction of the design matrices satisfying optimality conditions. The best general references here are Gail and Kiefer (1982), Ceranka and Graczyk (2010;2012), Katulska and Smaga (2010).
For any matrix we consider a linear model: , y = Xw + e (1) where y is an 1 n´ random vector of observed measurements. Moreover, w is a 1 p´ vector representing unknown measurements of objects and w is an 1 n´ vector of random errors. We make two standing assumptions: it is required that there are no systematic errors, i.e. E(e) = 0 n , and that the errors are uncorrelated and have different variances, i.e. Var(e) = 2 G s , where 0 s > is a known parameter, G is the n ń diagonal positive definite matrix of known elements.
For the estimation of the vector of unknown measurements of objects w, we use the normal equation . Under the assumption that G is a known positive definite matrix, 1 X'G X is nonsingular if and only if X is of full column rank. In the case when 1 X'G X is nonsingular, the generalised least squares estimator of w is given by The statistical problem considered here is how to determine the estimator of the vector of unknown measurements of objects w when the observations follow the model (1). Among several questions taken under consideration, the properties of this estimator are under considerations. The characteristic features are determined by the properties of the design. Especially here, it is expected that the product of the variance of the estimators has attained the lowest bound. Hence, the criterion of the D-optimality is considered. The design X D is D-optimal in the class of the designs It is known that The concept of D-optimality was considered in the books of Raghavarao (1971), Banerjee (1975), as well as Shah and Sinh (1989). Although theoretical studies on providing knowledge to guide the selection of optimal designs are not scarce, we are still unable to determine a regular D-optimal design for any combination of the number of objects and the number of measurements.
In such a case, a highly D-efficient design is considered. For details, we refer the reader to Bulutoglu and Ryan (2009). In Ceranka and Graczyk's (2018) paper, the definition of D-efficiency is given. We indicate a highly D-efficient design when where Y is the matrix of D-optimal spring balance weighing design. The aim of this paper is to develop new construction methods related to D-optimal and highly D-efficient spring balance weighing designs for which random errors are uncorrelated and have different variances. An attempt has been made here to expand the theory of optimal designs. The aim of this research is to develop the results concerning new methods of determining optimal designs in classes in which they have not been determined in the literature so far.

The main result
We present the theorem determining the parameters of the highly D-efficient design given in Ceranka and Graczyk (2018;2019).
Theorem 2.1. Let p be even. In any non-singular spring balance weighing design  having the form given in Theorem 2.1 is considered as highly D-efficient.

Addition of one measurement
be the design of the highly D-efficient spring balance weighing design. Now, let us consider the design where 1 x is any 1 p´ vector of elements 1 or 0, ' So, the variance matrix of errors is given as: Furthermore, we study the function Owing to the fact that ( ) In order to maximise the expression to the right of the inequality sign, we observe that ( ) ( ) The above-presented equality is fulfilled if and only if 1 2 2 is a highly D-efficient spring balance weighing design.
is a hhly D-efficient spring balance weighing design.

Addition of two measurements
be the design of the highly D-efficient spring balance weighing design. Now, let us consider the design where h x is a 1 p´ vector of elements 1 or 0, ' So, the variance matrix of errors is given as: Furthermore, we study the function In this situation, In this case, is a highly D-efficient spring balance weighing design.

Addition of three measurements
(2.6) In this case,  It is worth emphasising that spring balance weighing designs can be applied in all experiments in which the experimental factors are at two levels, see, for example, Ceranka and Graczyk (2014). Let us suppose that we study the real estate market and we are interested in the influence of the following factors: the prospect of further price increases in the local housing market, availability of loans for the purchase of apartments, the prospect of increasing VAT, fears related to the liquidation of the interest relief, the current price increase observed in the local market, availability of housing in the secondary market (each at two levels coded with 1 or 0). From the statistical point of view, we are interested in determining the influences of these factors using twenty different combinations. In the notation of weighing designs, we determine unknown measurements of p = 6 objects in n = 13 surveys, so we consider the class ( ) 13 6 0,1 Ψ . The scheme of determination of the measurements, i.e. the design matrix, is given in the example 2.6. Possible applications of the discussed designs should be sought wherever the measurement results can be written as a linear combination of unknown object measures with coefficients equal to 0 or 1. This paper was presented at the conference MSA 2019 which financed its publication. Organisation of the international conference "Multivariate Statistical Analysis 2019" (MSA 2019) was supported by resources for the popularisation of scientific activities of the Minister of Science and Higher Education in the framework of agreement No 712/PDUN/202019.